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1l55hphme | maths | straight-lines-and-pair-of-straight-lines | area-of-triangle-and-condition-of-collinearity | <p>Let a triangle be bounded by the lines L<sub>1</sub> : 2x + 5y = 10; L<sub>2</sub> : $$-$$4x + 3y = 12 and the line L<sub>3</sub>, which passes through the point P(2, 3), intersects L<sub>2</sub> at A and L<sub>1</sub> at B. If the point P divides the line-segment AB, internally in the ratio 1 : 3, then the area of ... | [{"identifier": "A", "content": "$${{110} \\over {13}}$$"}, {"identifier": "B", "content": "$${{132} \\over {13}}$$"}, {"identifier": "C", "content": "$${{142} \\over {13}}$$"}, {"identifier": "D", "content": "$${{151} \\over {13}}$$"}] | ["B"] | null | $L_{1}: 2 x+5 y=10$<br><br>
$L_{2}: -4 x+ 3 y=12$<br><br>
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l99tn43d/ebd74024-c280-415b-bc94-0859668a0f7c/89bb0690-4c7a-11ed-b94d-45a8040c2a81/file-1l99tn43e.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l99tn43d/ebd74024-c280-415b... | mcq | jee-main-2022-online-28th-june-evening-shift |
1l589p2la | maths | straight-lines-and-pair-of-straight-lines | area-of-triangle-and-condition-of-collinearity | <p>Let R be the point (3, 7) and let P and Q be two points on the line x + y = 5 such that PQR is an equilateral triangle. Then the area of $$\Delta$$PQR is :</p> | [{"identifier": "A", "content": "$${{25} \\over {4\\sqrt 3 }}$$"}, {"identifier": "B", "content": "$${{25\\sqrt 3 } \\over 2}$$"}, {"identifier": "C", "content": "$${{25} \\over {\\sqrt 3 }}$$"}, {"identifier": "D", "content": "$${{25} \\over {2\\sqrt 3 }}$$"}] | ["D"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l59j3d6d/8a4b3248-26d5-486b-9475-fbc8a7d508b1/6bc27690-fd20-11ec-8035-4341fba75e09/file-1l59j3d6h.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l59j3d6d/8a4b3248-26d5-486b-9475-fbc8a7d508b1/6bc27690-fd20-11ec-8035-4341fba75e09... | mcq | jee-main-2022-online-26th-june-morning-shift |
1l5b8adb6 | maths | straight-lines-and-pair-of-straight-lines | area-of-triangle-and-condition-of-collinearity | <p>Let the area of the triangle with vertices A(1, $$\alpha$$), B($$\alpha$$, 0) and C(0, $$\alpha$$) be 4 sq. units. If the points ($$\alpha$$, $$-$$$$\alpha$$), ($$-$$$$\alpha$$, $$\alpha$$) and ($$\alpha$$<sup>2</sup>, $$\beta$$) are collinear, then $$\beta$$ is equal to :</p> | [{"identifier": "A", "content": "64"}, {"identifier": "B", "content": "$$-$$8"}, {"identifier": "C", "content": "$$-$$64"}, {"identifier": "D", "content": "512"}] | ["C"] | null | <p>$$\because$$ A(1, $$\alpha$$), B($$\alpha$$, 0) and C(0, $$\alpha$$) are the vertices of $$\Delta$$ABC and area of $$\Delta$$ABC = 4</p>
<p>$$\therefore$$ $$\left| {{1 \over 2}\left| {\matrix{
1 & \alpha & 1 \cr
\alpha & 0 & 1 \cr
0 & \alpha & 1 \cr
} } \right|} \right| = 4$$</p>
<p>$$ \Rightarro... | mcq | jee-main-2022-online-24th-june-evening-shift |
1l5c29doe | maths | straight-lines-and-pair-of-straight-lines | area-of-triangle-and-condition-of-collinearity | <p>Let $$A\left( {{3 \over {\sqrt a }},\sqrt a } \right),\,a > 0$$, be a fixed point in the xy-plane. The image of A in y-axis be B and the image of B in x-axis be C. If $$D(3\cos \theta ,a\sin \theta )$$ is a point in the fourth quadrant such that the maximum area of $$\Delta$$ACD is 12 square units, then a is equa... | [] | null | 8 | Clearly $B$ is $\left(-\frac{3}{\sqrt{a}},+\sqrt{a}\right)$ and $C$ is $\left(-\frac{3}{\sqrt{a}},-\sqrt{a}\right)$
<br/><br/>
$$
\begin{aligned}
&\text { Area of } \triangle A C D=\frac{1}{2}\left|\begin{array}{ccc}
\frac{3}{\sqrt{a}} & \sqrt{a} & 1 \\\\
-\frac{3}{\sqrt{a}} & -\sqrt{a} & 1 \\\\
3 \cos \theta & a \sin ... | integer | jee-main-2022-online-24th-june-morning-shift |
1l5vzzg8x | maths | straight-lines-and-pair-of-straight-lines | area-of-triangle-and-condition-of-collinearity | <p>Let $$\alpha$$<sub>1</sub>, $$\alpha$$<sub>2</sub> ($$\alpha$$<sub>1</sub> < $$\alpha$$<sub>2</sub>) be the values of $$\alpha$$ fo the points ($$\alpha$$, $$-$$3), (2, 0) and (1, $$\alpha$$) to be collinear. Then the equation of the line, passing through ($$\alpha$$<sub>1</sub>, $$\alpha$$<sub>2</sub>) and makin... | [{"identifier": "A", "content": "$$x - \\sqrt 3 y - 3\\sqrt 3 + 1 = 0$$"}, {"identifier": "B", "content": "$$\\sqrt 3 x - y + \\sqrt 3 + 3 = 0$$"}, {"identifier": "C", "content": "$$x - \\sqrt 3 y + 3\\sqrt 3 + 1 = 0$$"}, {"identifier": "D", "content": "$$\\sqrt 3 x - y + \\sqrt 3 - 3 = 0$$"}] | ["B"] | null | <p>Points A($$\alpha$$, $$-$$3), B(2, 0) and C(1, $$\alpha$$) are collinear.</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5yktc03/7dba7f27-1a1c-4810-bc97-87e79ff18cdf/bf409530-0ae6-11ed-a51c-73986e88f75f/file-1l5yktc04.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l... | mcq | jee-main-2022-online-30th-june-morning-shift |
1l6jd47sy | maths | straight-lines-and-pair-of-straight-lines | area-of-triangle-and-condition-of-collinearity | <p>Let $$A(1,1), B(-4,3), C(-2,-5)$$ be vertices of a triangle $$A B C, P$$ be a point on side $$B C$$, and $$\Delta_{1}$$ and $$\Delta_{2}$$ be the areas of triangles $$A P B$$ and $$A B C$$, respectively. If $$\Delta_{1}: \Delta_{2}=4: 7$$, then the area enclosed by the lines $$A P, A C$$ and the $$x$$-axis is :</p> | [{"identifier": "A", "content": "$$\\frac{1}{4}$$"}, {"identifier": "B", "content": "$$\\frac{3}{4}$$"}, {"identifier": "C", "content": "$$\\frac{1}{2}$$"}, {"identifier": "D", "content": "1"}] | ["C"] | null | <p>$${{{\Delta _1}} \over {{\Delta _2}}} = {{{1 \over 2} \times BP \times AH} \over {{1 \over 2} \times BC \times AH}} = {4 \over 7}$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7psetoj/75213353-b6a6-48ee-b558-8da28250e29e/d5b7d130-2da9-11ed-8542-f96181a425b5/file-1l7psetok.png?format=png"... | mcq | jee-main-2022-online-27th-july-morning-shift |
1ldsu6789 | maths | straight-lines-and-pair-of-straight-lines | area-of-triangle-and-condition-of-collinearity | <p>Let $$B$$ and $$C$$ be the two points on the line $$y+x=0$$ such that $$B$$ and $$C$$ are symmetric with respect to the origin. Suppose $$A$$ is a point on $$y-2 x=2$$ such that $$\triangle A B C$$ is an equilateral triangle. Then, the area of the $$\triangle A B C$$ is :</p> | [{"identifier": "A", "content": "$$\\frac{10}{\\sqrt{3}}$$"}, {"identifier": "B", "content": "$$2 \\sqrt{3}$$"}, {"identifier": "C", "content": "$$3 \\sqrt{3}$$"}, {"identifier": "D", "content": "$$\\frac{8}{\\sqrt{3}}$$"}] | ["D"] | null | Origin $(O)$ is mid-point of $B C(x+y=0)$.
<br/><br/>
$A$ lies on perpendicular bisector of $B C$, which is $x-y=0$
<br/><br/>
A is point of intersection of $x-y=0$ and $y-2 x=2$
<br/><br/>
$\therefore A \equiv(-2,-2)$
<br/><br/>
Let $h=A O=\frac{-2-2}{\sqrt{1^{2}+1^{2}}}=2 \sqrt{2}$
<br/><br/>
$$
\text { Area }=\frac{... | mcq | jee-main-2023-online-29th-january-morning-shift |
luy6z4o4 | maths | straight-lines-and-pair-of-straight-lines | area-of-triangle-and-condition-of-collinearity | <p>A variable line $$\mathrm{L}$$ passes through the point $$(3,5)$$ and intersects the positive coordinate axes at the points $$\mathrm{A}$$ and $$\mathrm{B}$$. The minimum area of the triangle $$\mathrm{OAB}$$, where $$\mathrm{O}$$ is the origin, is :</p> | [{"identifier": "A", "content": "35"}, {"identifier": "B", "content": "25"}, {"identifier": "C", "content": "30"}, {"identifier": "D", "content": "40"}] | ["C"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lw3avn6c/d5ed79f8-059d-4be3-a5b7-c41490773adc/6f4ef630-103d-11ef-94ca-5b39b659c816/file-1lw3avn6d.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lw3avn6c/d5ed79f8-059d-4be3-a5b7-c41490773adc/6f4ef630-103d-11ef-94ca-5b39b659c816... | mcq | jee-main-2024-online-9th-april-morning-shift |
H01NW8Z2GN5a4Mxe | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | If a vertex of a triangle is $$(1, 1)$$ and the mid points of two sides through this vertex are $$(-1, 2)$$ and $$(3, 2)$$ then the centroid of the triangle is : | [{"identifier": "A", "content": "$$\\left( { - 1,{7 \\over 3}} \\right)$$ "}, {"identifier": "B", "content": "$$\\left( {{{ - 1} \\over 3},{7 \\over 3}} \\right)$$ "}, {"identifier": "C", "content": "$$\\left( { 1,{7 \\over 3}} \\right)$$"}, {"identifier": "D", "content": "$$\\left( {{{ 1} \\over 3},{7 \\over 3}} \\r... | ["C"] | null | Vertex of triangle is $$\left( {1,\,1} \right)$$ and midpoint of sides through -
<br><br>this vertex is $$\left( { - 1,\,2} \right)$$ and $$\left( {3,2} \right)$$
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263971/exam_images/d94fpagqlokdwur1hypq.webp" loading="lazy... | mcq | aieee-2005 |
OtQMa91jbhOp68Kd | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | The $$x$$-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as $$(0, 1) (1, 1)$$ and $$(1, 0)$$ is : | [{"identifier": "A", "content": "$$2 + \\sqrt 2 $$ "}, {"identifier": "B", "content": "$$2 - \\sqrt 2 $$"}, {"identifier": "C", "content": "$$1 + \\sqrt 2 $$"}, {"identifier": "D", "content": "$$1 - \\sqrt 2 $$"}] | ["B"] | null | From the figure, we have
<br><br>$$a = 2,b = 2\sqrt 2 ,c = 2$$
<br><br>$${x_1} = 0,\,{x^2} = 0,\,{x_3} = 2$$
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264731/exam_images/mfh7nksrzqack2i9ykdk.webp" loading="lazy" alt="JEE Main 2013 (Offline) Mathematics - Straight L... | mcq | jee-main-2013-offline |
JQXASfRf5qjuZV5Y | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | Let k be an integer such that the triangle with vertices (k, – 3k), (5, k) and (–k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point : | [{"identifier": "A", "content": "$$\\left( {1,{3 \\over 4}} \\right)$$"}, {"identifier": "B", "content": "$$\\left( {1, - {3 \\over 4}} \\right)$$"}, {"identifier": "C", "content": "$$\\left( {2,{1 \\over 2}} \\right)$$"}, {"identifier": "D", "content": "$$\\left( {2, - {1 \\over 2}} \\right)$$"}] | ["C"] | null | Given, vertices of triangle are (k, – 3k), (5, k) and (–k, 2).
<br><br>$${1 \over 2}\left| {\matrix{
k & { - 3k} & 1 \cr
5 & k & 1 \cr
{ - k} & 2 & 1 \cr
} } \right| = \pm 28$$
<br><br>$$ \Rightarrow $$ k(k - 2) + 3k(5 + k) + 1(10 + k<sup>2</sup>) = $$ \pm $$ 56
<br><br>$$ \Rig... | mcq | jee-main-2017-offline |
oKvo7JHtrCRqrm0obaHah | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | Let the equations of two sides of a triangle be 3x $$-$$ 2y + 6 = 0 and 4x + 5y $$-$$ 20 = 0. If the orthocentre of this triangle is at (1, 1), then the equation of its third side is : | [{"identifier": "A", "content": "122y $$-$$ 26x $$-$$ 1675 = 0"}, {"identifier": "B", "content": "122y + 26x + 1675 = 0"}, {"identifier": "C", "content": "26x + 61y + 1675 = 0"}, {"identifier": "D", "content": "26x $$-$$ 122y $$-$$ 1675 = 0"}] | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264905/exam_images/ubp63holkjbubzmm73xt.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 9th January Evening Slot Mathematics - Straight Lines and Pair of Straight Lines Question 104 En... | mcq | jee-main-2019-online-9th-january-evening-slot |
jzTjddlv24Pe3XwUGoOyx | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | A point P moves on the line 2x – 3y + 4 = 0. If Q(1, 4) and R (3, – 2) are fixed points, then the locus of the centroid of $$\Delta $$PQR is a line : | [{"identifier": "A", "content": "parallel to y-axis"}, {"identifier": "B", "content": "with slope $${2 \\over 3}$$"}, {"identifier": "C", "content": "parallel to x-axis"}, {"identifier": "D", "content": "with slope $${3 \\over 2}$$"}] | ["B"] | null | Let the centroid of $$\Delta $$PQR is (h, k) & P is ($$\alpha $$, $$\beta $$), then
<br><br>$${{\alpha + 1 + 3} \over 3} = h\,$$ and $${{\beta + 4 - 2} \over 3} = k$$
<br><br>$$\alpha = \left( {3h - 4} \right)$$ $$\beta = \left( {3k - 4} \right)$$
<br><br>Poin... | mcq | jee-main-2019-online-10th-january-morning-slot |
tkJLUql0mThR9SD5j5Xie | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | If the line 3x + 4y – 24 = 0 intersects the x-axis at the point A and the y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is : | [{"identifier": "A", "content": "(3, 4)"}, {"identifier": "B", "content": "(2, 2)"}, {"identifier": "C", "content": "(4, 4)"}, {"identifier": "D", "content": "(4, 3)"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265477/exam_images/migtvndabwdpqcwl9h0o.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 10th January Morning Slot Mathematics - Straight Lines and Pair of Straight Lines Question 101 E... | mcq | jee-main-2019-online-10th-january-morning-slot |
TWc5X8nsXEQEf4RxhueyF | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | Two vertices of a triangle are (0, 2) and (4, 3). If its orthocenter is at the origin, then its third vertex lies in which quadrant : | [{"identifier": "A", "content": "third"}, {"identifier": "B", "content": "fourth"}, {"identifier": "C", "content": "second"}, {"identifier": "D", "content": "first"}] | ["C"] | null | m<sub>BD</sub> $$ \times $$ m<sub>AD</sub> = $$-$$ 1
<br><br>$$ \Rightarrow $$ $$\left( {{{3 - 2} \over {4 - 0}}} \right) \times \left( {{{b - 0} \over {a - 0}}} \right) = - 1$$
<br><br>$$ \Rightarrow $$ b + 4a = 0 . . . . (i)
<br><br><img src="https://res.cloudinary... | mcq | jee-main-2019-online-10th-january-evening-slot |
8T16OEh0f3XuKhA4lw7k9k2k5e4orbt | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | Let A(1, 0), B(6, 2) and C $$\left( {{3 \over 2},6} \right)$$ be the vertices of a triangle ABC. If P is a Point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment PQ, where Q is the point $$\left( { - {7 \over 6}, - {1 \over 3}} \right)$$, is ________... | [] | null | 5 | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265627/exam_images/uw3nb7jjk16aarugxdjz.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 7th January Morning Slot Mathematics - Straight Lines and Pair of Straight Lines Question 86 Engli... | integer | jee-main-2020-online-7th-january-morning-slot |
kAbgo0MsBNIlywerYv7k9k2k5itigqx | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | Let C be the centroid of the triangle with
vertices (3, –1), (1, 3) and (2, 4). Let P be the
point of intersection of the lines x + 3y – 1 = 0
and 3x – y + 1 = 0. Then the line passing through
the points C and P also passes through the
point : | [{"identifier": "A", "content": "(\u20139, \u20137)"}, {"identifier": "B", "content": "(9, 7)"}, {"identifier": "C", "content": "(7, 6)"}, {"identifier": "D", "content": "(\u20139, \u20136)"}] | ["D"] | null | Centroid C $$\left( {{{3 + 1 + 2} \over 3},{{ - 1 + 3 + 4} \over 3}} \right)$$ = (2, 2)
<br><br>Point of intersection of lines x + 3y – 1 = 0 <br><br>and 3x – y + 1 = 0 is P $$\left( { - {1 \over 5},{2 \over 5}} \right)$$
<br><br>So, equation of line CP is 8x – 11y + 6 = 0
<br><br>Point (–9, –6) satisfy this equation. | mcq | jee-main-2020-online-9th-january-morning-slot |
9r0ym1rWgH3JCykDoIjgy2xukf4662pc | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | If a $$\Delta $$ABC has vertices A(–1, 7), B(–7, 1) and
C(5, –5), then its orthocentre has coordinates : | [{"identifier": "A", "content": "(\u20133, 3)"}, {"identifier": "B", "content": "(3, \u20133)"}, {"identifier": "C", "content": "$$\\left( {{3 \\over 5}, - {3 \\over 5}} \\right)$$"}, {"identifier": "D", "content": "$$\\left( { - {3 \\over 5},{3 \\over 5}} \\right)$$"}] | ["A"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267134/exam_images/ubvfpgl2agbziakghok1.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265216/exam_images/cip9scai14rag6uy6gdh.webp"><img src="https://res.c... | mcq | jee-main-2020-online-3rd-september-evening-slot |
2l8AfZuROD0j15pyvo1kmja51tj | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | In a triangle PQR, the co-ordinates of the points P and Q are ($$-$$2, 4) and (4, $$-$$2) respectively. If the equation of the perpendicular bisector of PR is 2x $$-$$ y + 2 = 0, then the centre of the circumcircle of the $$\Delta$$PQR is : | [{"identifier": "A", "content": "($$-$$1, 0)"}, {"identifier": "B", "content": "(1, 4)"}, {"identifier": "C", "content": "(0, 2)"}, {"identifier": "D", "content": "($$-$$2, $$-$$2)"}] | ["D"] | null | Mid point of $$PQ \equiv \left( {{{ - 2 + 4} \over 2},{{4 - 2} \over 2}} \right) \equiv (1,1)$$<br><br>Slope of $$PQ = {{4 + 2} \over { - 2 - 4}} = - 1$$<br><br>Slope of perpendicular bisector of PQ = 1<br><br>Equation of perpendicular bisector of PQ <br><br>$$y - 1 = 1(x - 1)$$<br><br>$$ \Rightarrow y = x$$<br><br>So... | mcq | jee-main-2021-online-17th-march-morning-shift |
0EJF9pQuqeuwViMpNb1kmknr7sp | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | Let tan$$\alpha$$, tan$$\beta$$ and tan$$\gamma$$; $$\alpha$$, $$\beta$$, $$\gamma$$ $$\ne$$ $${{(2n - 1)\pi } \over 2}$$, n$$\in$$N be the slopes of three line segments OA, OB and OC, respectively, where O is origin. If circumcentre of $$\Delta$$ABC coincides with origin and its orthocentre lies on y-axis, then the va... | [] | null | 144 | Since orthocentre and circumcentre both lies on y-axis.<br><br>$$ \Rightarrow $$ Centroid also lies on y-axis.<br><br>$$ \Rightarrow $$ $$\sum {\cos \alpha = 0} $$<br><br>cos$$\alpha$$ + cos$$\beta$$ + cos$$\gamma$$ = 0<br><br>$$ \Rightarrow $$ cos<sup>3</sup> $$\alpha$$ + cos<sup>3</sup> $$\beta$$ + cos<sup>3</sup> $... | integer | jee-main-2021-online-17th-march-evening-shift |
Qjcr2pxjkzDbBSuAFH1kmm3hfre | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x + y = 3. If R and r be the radius of circumcircle and incircle respectively of $$\Delta$$ABC, then (R + r) is equal to : | [{"identifier": "A", "content": "$$7\\sqrt 2 $$"}, {"identifier": "B", "content": "$${9 \\over {\\sqrt 2 }}$$"}, {"identifier": "C", "content": "$$2\\sqrt 2 $$"}, {"identifier": "D", "content": "$$3\\sqrt 2 $$"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264797/exam_images/ajzrnczcpsrafxwpsdbk.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 18th March Evening Shift Mathematics - Straight Lines and Pair of Straight Lines Question 65 Engli... | mcq | jee-main-2021-online-18th-march-evening-shift |
1krrwe4qx | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | Consider a triangle having vertices A($$-$$2, 3), B(1, 9) and C(3, 8). If a line L passing through the circum-centre of triangle ABC, bisects line BC, and intersects y-axis at point $$\left( {0,{\alpha \over 2}} \right)$$, then the value of real number $$\alpha$$ is ________________. | [] | null | 9 | <br>$${\left( {\sqrt {50} } \right)^2} = {\left( {\sqrt {45} } \right)^2} + {\left( {\sqrt 5 } \right)^2}$$<br><br>$$\angle B = 90^\circ $$<br><br>Circum-center $$ = \left( {{1 \over 2},{{11} \over 2}} \right)$$<br><br>Mid point of BC $$ = \left( {2,{{17} \over 2}} \right)$$<br><br>Line : $$\left( {y - {{11} \over 2}} ... | integer | jee-main-2021-online-20th-july-evening-shift |
1l54bfz5l | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>The distance of the origin from the centroid of the triangle whose two sides have the equations $$x - 2y + 1 = 0$$ and $$2x - y - 1 = 0$$ and whose orthocenter is $$\left( {{7 \over 3},{7 \over 3}} \right)$$ is :</p> | [{"identifier": "A", "content": "$$\\sqrt 2 $$"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "2$$\\sqrt 2 $$"}, {"identifier": "D", "content": "4"}] | ["C"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5gmnar6/e2444175-9dd2-4655-a827-d624ec162bae/ce00a120-0107-11ed-a5c5-a7461ac77783/file-1l5gmnar7.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5gmnar6/e2444175-9dd2-4655-a827-d624ec162bae/ce00a120-0107-11ed-a5c5-a7461ac77783... | mcq | jee-main-2022-online-29th-june-evening-shift |
1l57oeszr | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>In an isosceles triangle ABC, the vertex A is (6, 1) and the equation of the base BC is 2x + y = 4. Let the point B lie on the line x + 3y = 7. If ($$\alpha$$, $$\beta$$) is the centroid of $$\Delta$$ABC, then 15($$\alpha$$ + $$\beta$$) is equal to :</p> | [{"identifier": "A", "content": "39"}, {"identifier": "B", "content": "41"}, {"identifier": "C", "content": "51"}, {"identifier": "D", "content": "63"}] | ["C"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5q9xuo0/da6e885c-cdde-4480-b9d1-c22df8a701e5/f34a6800-0655-11ed-903e-c9687588b3f3/file-1l5q9xuo1.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5q9xuo0/da6e885c-cdde-4480-b9d1-c22df8a701e5/f34a6800-0655-11ed-903e-c9687588b3f3... | mcq | jee-main-2022-online-27th-june-morning-shift |
1l6gjlh74 | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>The equations of the sides $$\mathrm{AB}, \mathrm{BC}$$ and $$\mathrm{CA}$$ of a triangle $$\mathrm{ABC}$$ are $$2 x+y=0, x+\mathrm{p} y=15 \mathrm{a}$$ and $$x-y=3$$ respectively. If its orthocentre is $$(2, a),-\frac{1}{2}<\mathrm{a}<2$$, then $$\mathrm{p}$$ is equal to ______________.</p> | [] | null | 3 | <p> <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7nbzt6e/130dc0b4-c7c5-40cb-813c-5bded9b079b4/10d39160-2c50-11ed-9dc0-a1792fcc650d/file-1l7nbzt6f.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7nbzt6e/130dc0b4-c7c5-40cb-813c-5bded9b079b4/10d39160-2c50-11ed-9dc0-a1792fcc650... | integer | jee-main-2022-online-26th-july-morning-shift |
1l6kk3vb7 | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>The equations of the sides $$\mathrm{AB}, \mathrm{BC}$$ and CA of a triangle ABC are $$2 x+y=0, x+\mathrm{p} y=39$$ and $$x-y=3$$ respectively and $$\mathrm{P}(2,3)$$ is its circumcentre. Then which of the following is NOT true?</p> | [{"identifier": "A", "content": "$$(\\mathrm{AC})^{2}=9 \\mathrm{p}$$"}, {"identifier": "B", "content": "$$(\\mathrm{AC})^{2}+\\mathrm{p}^{2}=136$$"}, {"identifier": "C", "content": "$$32<\\operatorname{area}\\,(\\Delta \\mathrm{ABC})<36$$"}, {"identifier": "D", "content": "$$34<\\operatorname{area}\\,(\\triangle \\mat... | ["D"] | null | <p>Intersection of $$2x + y = 0$$ and $$x - y = 3\,:\,A(1, - 2)$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7qa6xtq/139436a6-31d3-4607-a05c-66e2b0bc9d4d/5ec55de0-2def-11ed-a744-1fb8f3709cfa/file-1l7qa6xtr.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7qa6xtq/139... | mcq | jee-main-2022-online-27th-july-evening-shift |
1l6p2cpyz | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1,1). If the line AP intersects the line BC at the point Q$$\left(k_{1}, k_{2}\right)$$, then $$k_{1}+k_{2}$$ is equal to :</p> | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "$$\\frac{4}{7}$$"}, {"identifier": "C", "content": "$$\\frac{2}{7}$$"}, {"identifier": "D", "content": "4"}] | ["B"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7ssjo3c/b55f47a7-9a89-4d60-875b-d115e046f61f/b6651380-2f50-11ed-85dd-19dc023e9ad1/file-1l7ssjo3d.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7ssjo3c/b55f47a7-9a89-4d60-875b-d115e046f61f/b6651380-2f50-11ed-85dd-19dc023e9ad1... | mcq | jee-main-2022-online-29th-july-morning-shift |
1l6relrm0 | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>Let $$\mathrm{A}(\alpha,-2), \mathrm{B}(\alpha, 6)$$ and $$\mathrm{C}\left(\frac{\alpha}{4},-2\right)$$ be vertices of a $$\triangle \mathrm{ABC}$$. If $$\left(5, \frac{\alpha}{4}\right)$$ is the circumcentre of $$\triangle \mathrm{ABC}$$, then which of the following is NOT correct about $$\triangle \mathrm{ABC}$$?<... | [{"identifier": "A", "content": "area is 24"}, {"identifier": "B", "content": "perimeter is 25"}, {"identifier": "C", "content": "circumradius is 5"}, {"identifier": "D", "content": "inradius is 2"}] | ["B"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7xrag3x/1171f510-f750-4a4d-8bf7-c934d96c9030/b8102bd0-320b-11ed-b25b-a342e8d3e498/file-1l7xrag3y.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7xrag3x/1171f510-f750-4a4d-8bf7-c934d96c9030/b8102bd0-320b-11ed-b25b-a342e8d3e498/fi... | mcq | jee-main-2022-online-29th-july-evening-shift |
1ldonjeqx | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>If the orthocentre of the triangle, whose vertices are (1, 2), (2, 3) and (3, 1) is $$(\alpha,\beta)$$, then the quadratic equation whose roots are $$\alpha+4\beta$$ and $$4\alpha+\beta$$, is :</p> | [{"identifier": "A", "content": "$$x^2-20x+99=0$$"}, {"identifier": "B", "content": "$$x^2-22x+120=0$$"}, {"identifier": "C", "content": "$$x^2-19x+90=0$$"}, {"identifier": "D", "content": "$$x^2-18x+80=0$$"}] | ["A"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lf41l718/e00587c4-d3cd-4c76-ac3c-9d17149d6abd/35cb41c0-c016-11ed-89c7-f9c9b186ec3c/file-1lf41l719.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lf41l718/e00587c4-d3cd-4c76-ac3c-9d17149d6abd/35cb41c0-c016-11ed-89c7-f9c9b186ec3c/fi... | mcq | jee-main-2023-online-1st-february-morning-shift |
1ldwxnvl2 | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>The equations of the sides AB, BC and CA of a triangle ABC are : $$2x+y=0,x+py=21a,(a\pm0)$$ and $$x-y=3$$ respectively. Let P(2, a) be the centroid of $$\Delta$$ABC. Then (BC)$$^2$$ is equal to ___________.</p> | [] | null | 122 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1le5hxc6e/38f5f2ee-c758-49c2-a1e2-99e79ae87394/aad05860-ad16-11ed-8a8c-4d67f5492755/file-1le5hxc6f.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1le5hxc6e/38f5f2ee-c758-49c2-a1e2-99e79ae87394/aad05860-ad16-11ed-8a8c-4d67f5492755... | integer | jee-main-2023-online-24th-january-evening-shift |
lgny7des | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | If $(\alpha, \beta)$ is the orthocenter of the triangle $\mathrm{ABC}$ with vertices $A(3,-7), B(-1,2)$ and $C(4,5)$, then $9 \alpha-6 \beta+60$ is equal to : | [{"identifier": "A", "content": "30"}, {"identifier": "B", "content": "40"}, {"identifier": "C", "content": "25"}, {"identifier": "D", "content": "35"}] | ["C"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lgxuayzx/c68e4c1e-f2ed-4189-96d3-77fa5b78bc40/7473b5d0-e445-11ed-97cc-4f9b4bf32610/file-1lgxuayzy.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lgxuayzx/c68e4c1e-f2ed-4189-96d3-77fa5b78bc40/7473b5d0-e445-11ed-97cc-4f9b4bf32610/fi... | mcq | jee-main-2023-online-15th-april-morning-shift |
1lgowjxyl | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>Let $$(\alpha, \beta)$$ be the centroid of the triangle formed by the lines $$15 x-y=82,6 x-5 y=-4$$ and $$9 x+4 y=17$$. Then $$\alpha+2 \beta$$ and $$2 \alpha-\beta$$ are the roots of the equation :</p> | [{"identifier": "A", "content": "$$x^{2}-7 x+12=0$$"}, {"identifier": "B", "content": "$$x^{2}-13 x+42=0$$"}, {"identifier": "C", "content": "$$x^{2}-14 x+48=0$$"}, {"identifier": "D", "content": "$$x^{2}-10 x+25=0$$"}] | ["B"] | null | <ol>
<li>Solve the equations $15x - y = 82$ and $6x - 5y = -4$</li>
</ol>
<p>Multiply the first equation by 5 and the second by 1 and then subtract the second from the first:</p>
<p>$75x - 5y = 410$</p>
<p>$6x - 5y = -4$</p>
<p>Subtracting these gives $69x = 414$ which leads to $x = 6$</p>
<p>Substitute $x = 6$ into th... | mcq | jee-main-2023-online-13th-april-evening-shift |
1lgzzli1e | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>Let $$C(\alpha, \beta)$$ be the circumcenter of the triangle formed by the lines</p>
<p>$$4 x+3 y=69$$</p>
<p>$$4 y-3 x=17$$, and</p>
<p>$$x+7 y=61$$.</p>
<p>Then $$(\alpha-\beta)^{2}+\alpha+\beta$$ is equal to :</p> | [{"identifier": "A", "content": "15"}, {"identifier": "B", "content": "17"}, {"identifier": "C", "content": "16"}, {"identifier": "D", "content": "18"}] | ["B"] | null | We have,
<br><br>$$
\begin{aligned}
& 4 x+3 y=69 .......(i) \\\\
& 4 y-3 x=17 .......(ii) \\\\
& x+7 y=61 .......(iii)
\end{aligned}
$$
<br><br>On solving (i) and (iii), we get,
<br><br>$$
\begin{aligned}
& x=12, \text { and } y=7 \\\\
& \text { So, } A \equiv(12,7)
\end{aligned}
$$
<br><br><img src... | mcq | jee-main-2023-online-8th-april-morning-shift |
lv5gs2w8 | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>If the orthocentre of the triangle formed by the lines $$2 x+3 y-1=0, x+2 y-1=0$$ and $$a x+b y-1=0$$, is the centroid of another triangle, whose circumcentre and orthocentre respectively are $$(3,4)$$ and $$(-6,-8)$$, then the value of $$|a-b|$$ is _________.</p> | [] | null | 16 | <p>Let circumcentre, orthocentre and centroid of a triangle $$P Q R$$ are $$C_1, H_1$$ and $$G_1$$ respectively</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lw8vj8jo/f80a9888-77a1-4003-a6a8-8ff4c790fc36/21421350-134e-11ef-8bbe-1b4949638519/file-1lw8vj8jp.png?format=png" data-orsrc="https://ap... | integer | jee-main-2024-online-8th-april-morning-shift |
qEqKSCymL9HY964D | maths | straight-lines-and-pair-of-straight-lines | distance-formula | A triangle with vertices $$\left( {4,0} \right),\left( { - 1, - 1} \right),\left( {3,5} \right)$$ is : | [{"identifier": "A", "content": "isosceles and right angled"}, {"identifier": "B", "content": "isosceles but not right angled"}, {"identifier": "C", "content": "right angled but not isosceles "}, {"identifier": "D", "content": "neither right angled nor isosceles "}] | ["A"] | null | $$AB = \sqrt {{{\left( {4 + 1} \right)}^2} + {{\left( {0 + 1} \right)}^2}} = \sqrt {26} ;$$
<br><br>$$BC = \sqrt {{{\left( {3 + 1} \right)}^2} + {{\left( {5 + 1} \right)}^2}} = \sqrt {52} $$
<br><br>$$CA = \sqrt {{{\left( {4 - 3} \right)}^2} + {{\left( {0 - 5} \right)}^2}} = \sqrt {26} ;$$
<br><br>In isosceles trian... | mcq | aieee-2002 |
1ktocacrw | maths | straight-lines-and-pair-of-straight-lines | distance-formula | A man starts walking from the point P($$-$$3, 4), touches the x-axis at R, and then turns to reach at the point Q(0, 2). The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then $$50\left( {{{(PR)}^2} + {{(RQ)}^2}} \right)$$ is equal to ____________. | [] | null | 1250 | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1kwjjmzcw/b86a6d68-7b5e-434f-a89c-eef8431d96b0/715950f0-5074-11ec-8a49-997c456a58f9/file-1kwjjmzcx.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1kwjjmzcw/b86a6d68-7b5e-434f-a89c-eef8431d96b0/715950f0-5074-11ec-8a49-997c456a58f9/fi... | integer | jee-main-2021-online-1st-september-evening-shift |
1l545buee | maths | straight-lines-and-pair-of-straight-lines | distance-formula | <p>The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A' B (where B is the point (2, 3)) subtend angle $${\pi \over 4}$$ at the origin, is equal to :</p> | [{"identifier": "A", "content": "10"}, {"identifier": "B", "content": "$${48 \\over 5}$$"}, {"identifier": "C", "content": "$${52 \\over 5}$$"}, {"identifier": "D", "content": "3"}] | ["C"] | null | Let $A(\alpha, 2) \quad$ Given $B(2,3)$
<br/><br/>
$$
\begin{aligned}
& m_{O A}=\frac{2}{\alpha} \quad\&\quad m_{O B}=\frac{3}{2} \\\\
& \tan \frac{\pi}{4}=\left|\frac{\frac{2}{\alpha}-\frac{3}{2}}{1+\frac{2}{\alpha} \cdot \frac{3}{2}}\right| \Rightarrow \frac{4-3 \alpha}{2 \alpha+6}=\pm 1 \\\\
& 4-3 \alpha=2 \alpha+6 ... | mcq | jee-main-2022-online-29th-june-morning-shift |
gJzbydMWci067R88 | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | The shortest distance between the line $$y - x = 1$$ and the curve $$x = {y^2}$$ is : | [{"identifier": "A", "content": "$${{2\\sqrt 3 } \\over 8}$$ "}, {"identifier": "B", "content": "$${{3\\sqrt 2 } \\over 5}$$"}, {"identifier": "C", "content": "$${{\\sqrt 3 } \\over 4}$$ "}, {"identifier": "D", "content": "$${{3\\sqrt 2 } \\over 8}$$"}] | ["D"] | null | Let $$\left( {{a^2},a} \right)$$ be the point of shortest distance on $$x = {y^2}$$
<br><br>Then distance between $$\left( {{a^2},a} \right)$$ and line $$x - y + 1 = 0$$
<br><br>is given by
<br><br>$$\,\,\,\,\,\,\,\,D = {{{a^2} - a + 1} \over {\sqrt 2 }} = {1 \over {\sqrt 2 }}\left[ {{{\left( {a - {1 \over 2}} \right)}... | mcq | aieee-2009 |
EXLH2cD628vv64gu | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | The line $$L$$ given by $${x \over 5} + {y \over b} = 1$$ passes through the point $$\left( {13,32} \right)$$. The line K is parrallel to $$L$$ and has the equation $${x \over c} + {y \over 3} = 1.$$ Then the distance between $$L$$ and $$K$$ is : | [{"identifier": "A", "content": "$$\\sqrt {17} $$ "}, {"identifier": "B", "content": "$${{17} \\over {\\sqrt {15} }}$$ "}, {"identifier": "C", "content": "$${{23} \\over {\\sqrt {17} }}$$ "}, {"identifier": "D", "content": "$${{23} \\over {\\sqrt {15} }}$$ "}] | ["C"] | null | Slope of line $$L = - {b \over 5}$$
<br><br>Slope of line $$K = - {3 \over c}$$
<br><br>Line $$L$$ is parallel to line $$k.$$
<br><br>$$ \Rightarrow {b \over 5} = {3 \over c} \Rightarrow bc = 15$$
<br><br>$$(13,32)$$ is a point on $$L.$$
<br><br>$$\therefore$$ $${{13} \over 5} + {{32} \over b} = 1 \Rightarrow {{32}... | mcq | aieee-2010 |
0czF19Zrf3iad8lsIiqYU | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | The foot of the perpendicular drawn from the origin, on the line, 3x + y = $$\lambda $$ ($$\lambda $$ $$ \ne $$ 0) is P. If the line meets x-axis at A and y-axis at B, then the ratio BP : PA is : | [{"identifier": "A", "content": "1 : 3"}, {"identifier": "B", "content": "3 : 1"}, {"identifier": "C", "content": "1 : 9"}, {"identifier": "D", "content": "9 : 1"}] | ["D"] | null | Equation of the line, which is perpendicular to the line,
<br><br>3x + y = $$\lambda $$($$\lambda $$ $$ \ne $$0) and passing through origin ,
<br><br>is given by $${{x - 0} \over 3} = {{y - 0} \over 1} = r$$
<br><br>For foot of perpendicular
<br><br>r = $${{ - \left( {\left( {3 \times 0} \right) + \left( {1 \times... | mcq | jee-main-2018-online-15th-april-evening-slot |
juSWMIEgIH0JUKssdD3rsa0w2w9jx2ezrxa | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | Lines are drawn parallel to the line 4x – 3y + 2 = 0, at a distance
$${3 \over 5}$$
from the origin. Then which one of the
following points lies on any of these lines ? | [{"identifier": "A", "content": "$$\\left( {{1 \\over 4}, - {1 \\over 3}} \\right)$$"}, {"identifier": "B", "content": "$$\\left( { - {1 \\over 4},{2 \\over 3}} \\right)$$"}, {"identifier": "C", "content": "$$\\left( { - {1 \\over 4}, - {2 \\over 3}} \\right)$$"}, {"identifier": "D", "content": "$$\\left( {{1 \\over 4}... | ["B"] | null | Line parallel to 4x – 3y + 2 = 0<br><br>
is given as 4x – 3y + $$\lambda $$ = 0<br><br>
distance from origin is <br><br>
$$\left| {{\lambda \over 5}} \right| = {3 \over 5}$$<br><br>
$$ \Rightarrow \lambda = \pm 3$$<br><br>
$$ \therefore $$ required lines are 4x – 3y + 3 = 0 & 4x – 3y – 3 = 0<br><br>
By Putting $... | mcq | jee-main-2019-online-10th-april-evening-slot |
9jOfE5xmA53cL9ZKq9jgy2xukfjjzmeu | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | If the line, 2x - y + 3 = 0 is at a distance<br/> $${1 \over {\sqrt 5 }}$$
and $${2 \over {\sqrt 5 }}$$ from the lines 4x - 2y + $$\alpha $$ = 0 <br/>and 6x - 3y + $$\beta $$ = 0, respectively, then the sum of all possible values of $$\alpha $$ and $$\beta $$ is :
| [] | null | 30 | Apply distance between parallel line formula<br><br>$$4x - 2y + \alpha = 0$$<br><br>$$4x - 2y + 6 = 0$$<br><br>$$\left| {{{\alpha - 6} \over {25}}} \right| = {1 \over {55}}$$<br><br>$$|\alpha - 6|\, = 2 \Rightarrow \alpha = 8,4$$<br><br>sum = 12<br><br>Again <br><br>$$6x - 3y + \beta = 0$$<br><br>$$6x - 3y + 9 = 0... | integer | jee-main-2020-online-5th-september-morning-slot |
1ktiq1mtm | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | If p and q are the lengths of the perpendiculars from the origin on the lines,<br/><br/>x cosec $$\alpha$$ $$-$$ y sec $$\alpha$$ = k cot 2$$\alpha$$ and<br/><br/>x sin$$\alpha$$ + y cos$$\alpha$$ = k sin2$$\alpha$$<br/><br/>respectively, then k<sup>2</sup> is equal to : | [{"identifier": "A", "content": "4p<sup>2</sup> + q<sup>2</sup>"}, {"identifier": "B", "content": "2p<sup>2</sup> + q<sup>2</sup>"}, {"identifier": "C", "content": "p<sup>2</sup> + 2q<sup>2</sup>"}, {"identifier": "D", "content": "p<sup>2</sup> + 4q<sup>2</sup>"}] | ["A"] | null | First line is $${x \over {\sin \alpha }} - {y \over {\cos \alpha }} = {{k\cos 2\alpha } \over {\sin 2\alpha }}$$<br><br>$$ \Rightarrow x\cos \alpha - y\sin \alpha = {k \over 2}\cos 2\alpha $$ <br><br>$$ \Rightarrow p = \left| {{k \over 2}\cos \alpha } \right| \Rightarrow 2p = \left| {k\cos 2\alpha } \right|$$ .... (i... | mcq | jee-main-2021-online-31st-august-morning-shift |
1l6f18gvo | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | <p>Let the point $$P(\alpha, \beta)$$ be at a unit distance from each of the two lines $$L_{1}: 3 x-4 y+12=0$$, and $$L_{2}: 8 x+6 y+11=0$$. If $$P$$ lies below $$L_{1}$$ and above $${ }{L_{2}}$$, then $$100(\alpha+\beta)$$ is equal to :</p> | [{"identifier": "A", "content": "$$-$$14"}, {"identifier": "B", "content": "42"}, {"identifier": "C", "content": "$$-$$22"}, {"identifier": "D", "content": "14"}] | ["D"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7bukhhd/39a595d1-0f44-4de9-a640-89709cea02f6/4c50df10-25ff-11ed-9c74-c5a04899a045/file-1l7bukhhe.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7bukhhd/39a595d1-0f44-4de9-a640-89709cea02f6/4c50df10-25ff-11ed-9c74-c5a04899a045... | mcq | jee-main-2022-online-25th-july-evening-shift |
1lgvr0lnr | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | <p>Let the equations of two adjacent sides of a parallelogram $$\mathrm{ABCD}$$ be $$2 x-3 y=-23$$ and $$5 x+4 y=23$$. If the equation of its one diagonal $$\mathrm{AC}$$ is $$3 x+7 y=23$$ and the distance of A from the other diagonal is $$\mathrm{d}$$, then $$50 \mathrm{~d}^{2}$$ is equal to ____________.</p> | [] | null | 529 | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lnkfs1i7/86b3f2b6-ecef-40f3-8533-8213186f85af/f96ad2f0-677b-11ee-aeb9-a1910acfd49b/file-6y3zli1lnkfs1i8.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lnkfs1i7/86b3f2b6-ecef-40f3-8533-8213186f85af/f96ad2f0-677b-11ee-ae... | integer | jee-main-2023-online-10th-april-evening-shift |
jaoe38c1lscoky97 | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | <p>If the sum of squares of all real values of $$\alpha$$, for which the lines $$2 x-y+3=0,6 x+3 y+1=0$$ and $$\alpha x+2 y-2=0$$ do not form a triangle is $$p$$, then the greatest integer less than or equal to $$p$$ is _________.</p> | [] | null | 32 | <p>$$\begin{aligned}
& 2 x-y+3=0 \\
& 6 x+3 y+1=0 \\
& \alpha x+2 y-2=0
\end{aligned}$$</p>
<p>Will not form a $$\Delta$$ if $$\alpha x+2 y-2=0$$ is concurrent with $$2 x-y+3=0$$ and $$6 x+3 y+1=0$$ or parallel to either of them so</p>
<p>Case-1: Concurrent lines</p>
<p>$$\left|\begin{array}{ccc}
2 & -1 & 3 \\
6 & 3 & ... | integer | jee-main-2024-online-27th-january-evening-shift |
jaoe38c1lsd4u0pr | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | <p>Let $$A(a, b), B(3,4)$$ and $$C(-6,-8)$$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $$P(2 a+3,7 b+5)$$ from the line $$2 x+3 y-4=0$$ measured parallel to the line $$x-2 y-1=0$$ is</p> | [{"identifier": "A", "content": "$$\\frac{17 \\sqrt{5}}{6}$$\n"}, {"identifier": "B", "content": "$$\\frac{15 \\sqrt{5}}{7}$$\n"}, {"identifier": "C", "content": "$$\\frac{17 \\sqrt{5}}{7}$$\n"}, {"identifier": "D", "content": "$$\\frac{\\sqrt{5}}{17}$$"}] | ["C"] | null | <p>$$\mathrm{A}(\mathrm{a}, \mathrm{b}), \quad \mathrm{B}(3,4), \quad \mathrm{C}(-6,-8)$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsjwjvlw/dec041c9-e406-4f98-a94d-783370c1f55d/ddd44840-ca2d-11ee-8854-3b5a6c9e9092/file-6y3zli1lsjwjvlx.png?format=png" data-orsrc="https://app-content.... | mcq | jee-main-2024-online-31st-january-evening-shift |
jaoe38c1lsfkb2im | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | <p>Let $$\mathrm{A}$$ be the point of intersection of the lines $$3 x+2 y=14,5 x-y=6$$ and $$\mathrm{B}$$ be the point of intersection of the lines $$4 x+3 y=8,6 x+y=5$$. The distance of the point $$P(5,-2)$$ from the line $$\mathrm{AB}$$ is</p> | [{"identifier": "A", "content": "$$\\frac{13}{2}$$"}, {"identifier": "B", "content": "8"}, {"identifier": "C", "content": "$$\\frac{5}{2}$$"}, {"identifier": "D", "content": "6"}] | ["D"] | null | <p>Solving lines $$\mathrm{L}_1(3 \mathrm{x}+2 \mathrm{y}=14)$$ and $$\mathrm{L}_2(5 \mathrm{x}-\mathrm{y}=6)$$ to get $$\mathrm{A}(2,4)$$ and solving lines $$\mathrm{L}_3(4 \mathrm{x}+3 \mathrm{y}=8)$$ and $$\mathrm{L}_4(6 \mathrm{x}+\mathrm{y}=5)$$ to get $$\mathrm{B}\left(\frac{1}{2}, 2\right)$$.</p>
<p>Finding Eqn.... | mcq | jee-main-2024-online-29th-january-evening-shift |
jaoe38c1lsfkdr4d | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | <p>The distance of the point $$(2,3)$$ from the line $$2 x-3 y+28=0$$, measured parallel to the line $$\sqrt{3} x-y+1=0$$, is equal to</p> | [{"identifier": "A", "content": "$$3+4 \\sqrt{2}$$\n"}, {"identifier": "B", "content": "$$6 \\sqrt{3}$$\n"}, {"identifier": "C", "content": "$$4+6 \\sqrt{3}$$\n"}, {"identifier": "D", "content": "$$4 \\sqrt{2}$$"}] | ["C"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsr8s48w/7106c58a-521f-4564-b530-e2a9d2152dfd/2d124800-ce37-11ee-9412-cd4f9c6f2c40/file-6y3zli1lsr8s48x.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lsr8s48w/7106c58a-521f-4564-b530-e2a9d2152dfd/2d124800-ce37-11ee... | mcq | jee-main-2024-online-29th-january-evening-shift |
1lsg40lwf | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | <p>If $$x^2-y^2+2 h x y+2 g x+2 f y+c=0$$ is the locus of a point, which moves such that it is always equidistant from the lines $$x+2 y+7=0$$ and $$2 x-y+8=0$$, then the value of $$g+c+h-f$$ equals</p> | [{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "14"}, {"identifier": "C", "content": "29"}, {"identifier": "D", "content": "6"}] | ["B"] | null | <p>Cocus of point $$\mathrm{P}(\mathrm{x}, \mathrm{y})$$ whose distance from Gives
$$X+2 y+7=0$$ & $$2 x-y+8=0$$ are equal is $$\frac{x+2 y+7}{\sqrt{5}}= \pm \frac{2 x-y+8}{\sqrt{5}}$$</p>
<p>$$(x+2 y+7)^2-(2 x-y+8)^2=0$$</p>
<p>Combined equation of lines</p>
<p>$$\begin{aligned}
& (x-3 y+1)(3 x+y+15)=0 \\
& 3 x^2-3 y^... | mcq | jee-main-2024-online-30th-january-evening-shift |
lv0vxcqn | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | <p>The vertices of a triangle are $$\mathrm{A}(-1,3), \mathrm{B}(-2,2)$$ and $$\mathrm{C}(3,-1)$$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :</p> | [{"identifier": "A", "content": "$$-x+y-(2-\\sqrt{2})=0$$\n"}, {"identifier": "B", "content": "$$x+y-(2-\\sqrt{2})=0$$\n"}, {"identifier": "C", "content": "$$x+y+(2-\\sqrt{2})=0$$\n"}, {"identifier": "D", "content": "$$x-y-(2+\\sqrt{2})=0$$"}] | ["B"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwk4845t/9c9451b9-d46c-4c1c-ab08-287bac97c8b0/bd468810-197c-11ef-8e15-2107d8c35500/file-1lwk4845u.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lwk4845t/9c9451b9-d46c-4c1c-ab08-287bac97c8b0/bd468810-197c-11ef-8e15-2107d8c35500... | mcq | jee-main-2024-online-4th-april-morning-shift |
mxfnnIcoMHiIDWz92h6ya | maths | straight-lines-and-pair-of-straight-lines | family-of-straight-line | Consider the set of all lines px + qy + r = 0 such that 3p + 2q + 4r = 0. Which one of the following statements
is true?
| [{"identifier": "A", "content": "The lines are not concurrent"}, {"identifier": "B", "content": "The lines are concurrent at the point $$\\left( {{3 \\over 4},{1 \\over 2}} \\right)$$"}, {"identifier": "C", "content": "The lines are all parallel "}, {"identifier": "D", "content": "Each line passes through the origin"}... | ["B"] | null | Equation of lines;
<br><br>px + qy + r = 0 . . . . . (1)
<br><br>Also given
<br><br>3p + 2q + 4r = 0 . . . . . . (2)
<br><br>divide equation (2) by 4, we get
<br><br>$${3 \over 4}P + {2 \over 4}q + r = 0$$ . . . . (3)
<br><br>By comparing (1) and (3) we get,
<br><br>x = $${3 \over ... | mcq | jee-main-2019-online-9th-january-morning-slot |
1MTF5OuuQpiWHnCU | maths | straight-lines-and-pair-of-straight-lines | locus | Locus of mid point of the portion between the axes of
<br/><br/>$$x$$ $$cos$$ $$\alpha + y\,\sin \alpha = p$$ where $$p$$ is constant is : | [{"identifier": "A", "content": "$${x^2} + {y^2} = {4 \\over {{p^2}}}$$ "}, {"identifier": "B", "content": "$${x^2} + {y^2} = 4{p^2}$$ "}, {"identifier": "C", "content": "$${1 \\over {{x^2}}} + {1 \\over {{y^2}}} = {2 \\over {{p^2}}}$$ "}, {"identifier": "D", "content": "$${1 \\over {{x^2}}} + {1 \\over {{y^2}}} = {4 \... | ["D"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264686/exam_images/ug1vcgctwnpzuwlv3nqn.webp" loading="lazy" alt="AIEEE 2002 Mathematics - Straight Lines and Pair of Straight Lines Question 116 English Explanation">
<br><br>Equation of $$AB$$ is
<br><br>$$x\cos \alpha + y\sin \... | mcq | aieee-2002 |
2Jxdn4DG7BLAO7DQ | maths | straight-lines-and-pair-of-straight-lines | locus | Locus of centroid of the triangle whose vertices are $$\left( {a\cos t,a\sin t} \right),\left( {b\sin t, - b\cos t} \right)$$ and $$\left( {1,0} \right),$$ where $$t$$ is a parameter, is : | [{"identifier": "A", "content": "$${\\left( {3x + 1} \\right)^2} + {\\left( {3y} \\right)^2} = {a^2} - {b^2}$$ "}, {"identifier": "B", "content": "$${\\left( {3x - 1} \\right)^2} + {\\left( {3y} \\right)^2} = {a^2} - {b^2}$$"}, {"identifier": "C", "content": "$${\\left( {3x - 1} \\right)^2} + {\\left( {3y} \\right)^2} ... | ["C"] | null | $$x = {{a\cos t + b\sin t + 1} \over 3}$$
<br><br>$$ \Rightarrow a\cos t + b\sin t = 3x - 1$$
<br><br>$$y = {{a\sin t - b\cos t} \over 3}$$
<br><br>$$ \Rightarrow a\sin t - b\cos t = 3y$$ | mcq | aieee-2003 |
E8ErhhYaYvb3mkqG | maths | straight-lines-and-pair-of-straight-lines | locus | If the equation of the locus of a point equidistant from the point $$\left( {{a_{1,}}{b_1}} \right)$$ and $$\left( {{a_{2,}}{b_2}} \right)$$ is
<br/>$$\left( {{a_1} - {a_2}} \right)x + \left( {{b_1} - {b_2}} \right)y + c = 0$$ , then the value of $$'c'$$ is : | [{"identifier": "A", "content": "$$\\sqrt {{a_1}^2 + {b_1}^2 - {a_2}^2 - {b_2}^2} $$ "}, {"identifier": "B", "content": "$${1 \\over 2}\\left( {{a_2}^2 + {b_2}^2 - {a_1}^2 - {b_1}^2} \\right)$$ "}, {"identifier": "C", "content": "$${{a_1}^2 - {a_2}^2 + {b_1}^2 - {b_2}^2}$$ "}, {"identifier": "D", "content": "$${1 \\ove... | ["B"] | null | Since, the points $\left(a_1, b_1\right)$ and $\left(a_2, b_2\right)$ satisfy the equation. So, that
<br/><br/>$$
\begin{aligned}
& a_1\left(a_1-a_2\right)+b_1\left(b_1-b_2\right)+c=0 ~~........(1) \\\\
& \text { and } a_2\left(a_1-a_2\right)+b_2\left(b_1-b_2\right)+c=0 ~~.........(2) \\\\
& \text { On adding Eqs. (i) ... | mcq | aieee-2003 |
cq2gQ4UBsv796eLK | maths | straight-lines-and-pair-of-straight-lines | locus | Let $$A\left( {2, - 3} \right)$$ and $$B\left( {-2, 1} \right)$$ be vertices of a triangle $$ABC$$. If the centroid of this triangle moves on the line $$2x + 3y = 1$$, then the locus of the vertex $$C$$ is the line : | [{"identifier": "A", "content": "$$3x - 2y = 3$$"}, {"identifier": "B", "content": "$$2x - 3y = 7$$"}, {"identifier": "C", "content": "$$3x + 2y = 5$$"}, {"identifier": "D", "content": "$$2x + 3y = 9$$"}] | ["D"] | null | Let the vertex $$C$$ be $$(h,k),$$ then the
<br><br>centroid of $$\Delta ABC$$ is $$\left( {{{2 + (- 2) + h} \over 3},{{ - 3 + 1 + k} \over 3}} \right)$$
<br><br>or $$\left( {{h \over 3},{{ - 2 + k} \over 3}} \right).$$ It lies on $$2x+3y=1$$
<br><br>$$ \Rightarrow {{2h} \over 3} - 2 + k = 1$$
<br><br>$$ \Rightarro... | mcq | aieee-2004 |
2OQtj1HvFLf6RH1ewBHeh | maths | straight-lines-and-pair-of-straight-lines | locus | If a variable line drawn through the intersection of the lines $${x \over 3} + {y \over 4} = 1$$ and $${x \over 4} + {y \over 3} = 1,$$ meets the coordinate axes at A and B, (A $$ \ne $$ B), then the locus of the midpoint of AB is : | [{"identifier": "A", "content": "6xy = 7(x + y)"}, {"identifier": "B", "content": "4(x + y)<sup>2</sup> \u2212 28(x + y) + 49 = 0"}, {"identifier": "C", "content": "7xy = 6(x + y)\n"}, {"identifier": "D", "content": "14(x + y)<sup>2</sup> \u2212 97(x + y) + 168 = 0"}] | ["C"] | null | L<sub>1</sub> : 4x + 3y $$-$$ 12 = 0
<br><br>L<sub>2</sub> : 3x + 4y $$-$$ 12 = 0
<br><br>Equation of line passing through the intersection of these two lines L<sub>1</sub> and L<sub>2</sub> is
<br><br>L<sub>1</sub> + $$\lambda $$L<sub>2</sub> = 0
<br><br>$$ \Rightarrow $$$$\,\,\,$$(4x + 3y $$-$$ 12) + $$\lambda $$(3... | mcq | jee-main-2016-online-9th-april-morning-slot |
cQrKvrI7xZYjN5kt | maths | straight-lines-and-pair-of-straight-lines | locus | A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is
the origin and the rectangle OPRQ is completed, then the locus of R is : | [{"identifier": "A", "content": "3x + 2y = 6xy"}, {"identifier": "B", "content": "3x + 2y = 6"}, {"identifier": "C", "content": "2x + 3y = xy"}, {"identifier": "D", "content": "3x + 2y = xy"}] | ["D"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265351/exam_images/bpha9tfo2ojwynaftia1.webp" loading="lazy" alt="JEE Main 2018 (Offline) Mathematics - Straight Lines and Pair of Straight Lines Question 119 English Explanation">
<br><br>Let coordinate of point R = (h, k).
<br><b... | mcq | jee-main-2018-offline |
3BmWgWmgNz7RCJsg8h9jZ | maths | straight-lines-and-pair-of-straight-lines | locus | Let O(0, 0) and A(0, 1) be two fixed points. Then
the locus of a point P such that the perimeter of
$$\Delta $$AOP is 4, is : | [{"identifier": "A", "content": "9x<sup>2</sup> + 8y<sup>2</sup> \u2013 8y = 16\n"}, {"identifier": "B", "content": "8x<sup>2</sup> \u2013 9y<sup>2</sup> + 9y = 18"}, {"identifier": "C", "content": "8x<sup>2</sup> + 9y<sup>2</sup> \u2013 9y = 18\n"}, {"identifier": "D", "content": "9x<sup>2</sup> \u2013 8y<sup>2</sup> ... | ["A"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267479/exam_images/wjrxugubrcihnruos7c4.webp"><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265282/exam_images/uonftnxl941nxalen4tg.webp" style="max-width: 100%;height: auto;display: block;margi... | mcq | jee-main-2019-online-8th-april-morning-slot |
KozcYkC6ueHANPacpT7k9k2k5fmtcf6 | maths | straight-lines-and-pair-of-straight-lines | locus | The locus of the mid-points of the perpendiculars drawn from points on the line, x = 2y to the line
x = y is : | [{"identifier": "A", "content": "3x - 2y = 0"}, {"identifier": "B", "content": "7x - 5y = 0\n"}, {"identifier": "C", "content": "2x - 3y = 0"}, {"identifier": "D", "content": "5x - 7y = 0"}] | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263683/exam_images/w8qnmvdmubr7jibhtirj.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 7th January Evening Slot Mathematics - Straight Lines and Pair of Straight Lines Question 85 Engli... | mcq | jee-main-2020-online-7th-january-evening-slot |
fsvYqfrKekcQWsXuJU1kmlm4kfs | maths | straight-lines-and-pair-of-straight-lines | locus | A square ABCD has all its vertices on the curve x<sup>2</sup>y<sup>2</sup> = 1. The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is _________. | [] | null | 80 | x<sup>2</sup>y<sup>2</sup> = 1
<br><br>$$ \Rightarrow $$ y<sup>2</sup> = $${1 \over {{x^2}}}$$
<br><br>$$ \Rightarrow $$ y = $$ \pm {1 \over x}$$
<br><br>Graph of this equation,
<br><br>$$OA \bot OB$$<br><br>$$ \Rightarrow \left( {{1 \over {{p^2}}}} \right)\left( { - {1 \over {{q^2}}}} \right) = - 1$$<br><br>$$ \Right... | integer | jee-main-2021-online-18th-march-morning-shift |
1ktehxolh | maths | straight-lines-and-pair-of-straight-lines | locus | Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y-axis at C. The locus of the mid-point P of MC is : | [{"identifier": "A", "content": "3x<sup>2</sup> $$-$$ 2y $$-$$ 6 = 0"}, {"identifier": "B", "content": "3x<sup>2</sup> + 2y $$-$$ 6 = 0"}, {"identifier": "C", "content": "2x<sup>2</sup> + 3y $$-$$ 9 = 0"}, {"identifier": "D", "content": "2x<sup>2</sup> $$-$$ 3y + 9 = 0"}] | ["C"] | null | A(0, 6) and B(2t, 0)<br><br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265928/exam_images/t38wivroym0l1mk7udq7.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 27th August Morning Shift Mathematics - Straight Lines and Pair of Str... | mcq | jee-main-2021-online-27th-august-morning-shift |
1l6gijngm | maths | straight-lines-and-pair-of-straight-lines | locus | <p>A point $$P$$ moves so that the sum of squares of its distances from the points $$(1,2)$$ and $$(-2,1)$$ is 14. Let $$f(x, y)=0$$ be the locus of $$\mathrm{P}$$, which intersects the $$x$$-axis at the points $$\mathrm{A}$$, $$\mathrm{B}$$ and the $$y$$-axis at the points C, D. Then the area of the quadrilateral ACBD... | [{"identifier": "A", "content": "$${9 \\over 2}$$"}, {"identifier": "B", "content": "$${{3\\sqrt {17} } \\over 2}$$"}, {"identifier": "C", "content": "$${{3\\sqrt {17} } \\over 4}$$"}, {"identifier": "D", "content": "9"}] | ["B"] | null | <p>Let point $$P:(h,\,k)$$</p>
<p>$${(h - 1)^2} + {(k - 2)^2} + {(h + 2)^2} + {(k - 1)^2} = 14$$</p>
<p>$$2{h^2} + 2{k^2} + 2h - 6k - 4 = 0$$</p>
<p>Locus of $$P:{x^2} + {y^2} + x - 3y - 2 = 0$$</p>
<p>Intersection with x-axis,</p>
<p>$${x^2} + x - 2 = 0$$</p>
<p>$$ \Rightarrow x = - 2,\,1$$</p>
<p>Intersection with y... | mcq | jee-main-2022-online-26th-july-morning-shift |
1lgrg2uy5 | maths | straight-lines-and-pair-of-straight-lines | locus | <p>If the point $$\left(\alpha, \frac{7 \sqrt{3}}{3}\right)$$ lies on the curve traced by the mid-points of the line segments of the lines $$x \cos \theta+y \sin \theta=7, \theta \in\left(0, \frac{\pi}{2}\right)$$ between the co-ordinates axes, then $$\alpha$$ is equal to :</p> | [{"identifier": "A", "content": "$$-$$7"}, {"identifier": "B", "content": "7"}, {"identifier": "C", "content": "$$-$$7$$\\sqrt3$$"}, {"identifier": "D", "content": "7$$\\sqrt3$$"}] | ["B"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lhu7fnjs/31373c2a-3f37-47d1-bed5-e74cffaeb86b/e21d2070-f611-11ed-8eab-4b3832376002/file-1lhu7fnjt.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lhu7fnjs/31373c2a-3f37-47d1-bed5-e74cffaeb86b/e21d2070-f611-11ed-8eab-4b3832376002/fi... | mcq | jee-main-2023-online-12th-april-morning-shift |
lv9s201x | maths | straight-lines-and-pair-of-straight-lines | locus | <p>Let $$\mathrm{A}(-1,1)$$ and $$\mathrm{B}(2,3)$$ be two points and $$\mathrm{P}$$ be a variable point above the line $$\mathrm{AB}$$ such that the area of $$\triangle \mathrm{PAB}$$ is 10. If the locus of $$\mathrm{P}$$ is $$\mathrm{a} x+\mathrm{by}=15$$, then $$5 \mathrm{a}+2 \mathrm{~b}$$ is :</p> | [{"identifier": "A", "content": "$$-\\frac{12}{5}$$"}, {"identifier": "B", "content": "$$-\\frac{6}{5}$$"}, {"identifier": "C", "content": "6"}, {"identifier": "D", "content": "4"}] | ["A"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lvnxhk5n/61340bd7-15c2-4d49-8787-6b1d30c56e5c/4f3a81b0-07c9-11ef-afdb-fb69ed1077de/file-6y3zli1lvnxhk5o.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lvnxhk5n/61340bd7-15c2-4d49-8787-6b1d30c56e5c/4f3a81b0-07c9-11ef-af... | mcq | jee-main-2024-online-5th-april-evening-shift |
lvb294xq | maths | straight-lines-and-pair-of-straight-lines | locus | <p>If the locus of the point, whose distances from the point $$(2,1)$$ and $$(1,3)$$ are in the ratio $$5: 4$$, is $$a x^2+b y^2+c x y+d x+e y+170=0$$, then the value of $$a^2+2 b+3 c+4 d+e$$ is equal to :</p> | [{"identifier": "A", "content": "37"}, {"identifier": "B", "content": "$$-27$$"}, {"identifier": "C", "content": "437"}, {"identifier": "D", "content": "5"}] | ["A"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwagdtgz/adc2b178-39c7-4610-ae95-7cf96da59f71/72edac30-142c-11ef-983f-65b4bd1ca415/file-1lwagdth0.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lwagdtgz/adc2b178-39c7-4610-ae95-7cf96da59f71/72edac30-142c-11ef-983f-65b4bd1ca415... | mcq | jee-main-2024-online-6th-april-evening-shift |
OYKsgMxcEWV5KKqu | maths | straight-lines-and-pair-of-straight-lines | pair-of-straight-lines | If the pair of lines
<br/><br/>$$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$$
<br/><br/>intersect on the $$y$$-axis then : | [{"identifier": "A", "content": "$$2fgh = b{g^2} + c{h^2}$$ "}, {"identifier": "B", "content": "$$b{g^2} \\ne c{h^2}$$ "}, {"identifier": "C", "content": "$$abc = 2fgh$$ "}, {"identifier": "D", "content": "none of these "}] | ["A"] | null | Put $$x=0$$ in the given equation
<br><br>$$ \Rightarrow b{y^2} + 2fy + c = 0.$$
<br><br>For unique point of intersection $${f^2} - bc = 0$$
<br><br>$$ \Rightarrow a{f^2} - abc = 0.$$
<br><br>Since $$abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0$$
<br><br>$$ \Rightarrow 2fgh - b{g^2} - c{h^2} = 0$$ | mcq | aieee-2002 |
nRhgOiVMJ3weRKlW | maths | straight-lines-and-pair-of-straight-lines | pair-of-straight-lines | The pair of lines represented by
$$$3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0$$$
<br/><br/>are perpendicular to each other for : | [{"identifier": "A", "content": "two values of $$a$$"}, {"identifier": "B", "content": "$$\\forall \\,a$$ "}, {"identifier": "C", "content": "for one value of $$a$$ "}, {"identifier": "D", "content": "for no values of $$a$$ "}] | ["A"] | null | $$3a + {a^2} - 2 = 0 \Rightarrow {a^2} + 3a - 2 = 0;$$
<br><br>$$ \Rightarrow a = {{ - 3 \pm \sqrt {9 + 8} } \over 2} = {{ - 3 \pm \sqrt {17} } \over 2}$$ | mcq | aieee-2002 |
d4SHDHZK60aYSFFU | maths | straight-lines-and-pair-of-straight-lines | pair-of-straight-lines | If the pair of straight lines $${x^2} - 2pxy - {y^2} = 0$$ and $${x^2} - 2qxy - {y^2} = 0$$ be such that each pair bisects the angle between the other pair, then : | [{"identifier": "A", "content": "$$pq = -1$$ "}, {"identifier": "B", "content": "$$p = q$$ "}, {"identifier": "C", "content": "$$p = -q$$ "}, {"identifier": "D", "content": "$$pq = 1$$."}] | ["A"] | null | Equation of bisectors of second pair of straight lines is,
<br><br>$$q{x^2} + 2xy - q{y^2} = 0\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$
<br><br>It must be identical to the first pair
<br><br>$${x^2} - 2\,pxy - {y^2} = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left(... 2 \right)$$
<br><br>from $$(1)$$ and $$(2)$$ $${q ... | mcq | aieee-2003 |
LKV0RhPLyjQeTiXA | maths | straight-lines-and-pair-of-straight-lines | pair-of-straight-lines | If the sum of the slopes of the lines given by $${x^2} - 2cxy - 7{y^2} = 0$$ is four times their product $$c$$ has the value : | [{"identifier": "A", "content": "$$-2$$ "}, {"identifier": "B", "content": "$$-1$$ "}, {"identifier": "C", "content": "$$2$$ "}, {"identifier": "D", "content": "$$1$$"}] | ["C"] | null | Let the lines be $$y = {m_1}x$$ and $$y = {m_2}x$$ then
<br><br>$${m_1} + {m_2} = - {{2c} \over 7}$$ and $${m_1}{m_2} = - {1 \over 7}$$
<br><br>Given $${m_1} + {m_2} = 4m{}_1{m_2}$$
<br><br>$$ \Rightarrow {{2c} \over 7} = - {4 \over 7} \Rightarrow c = 2$$ | mcq | aieee-2004 |
Cy25MQi1BYWBv2tA | maths | straight-lines-and-pair-of-straight-lines | pair-of-straight-lines | If one of the lines given by $$6{x^2} - xy + 4c{y^2} = 0$$ is $$3x + 4y = 0,$$ then $$c$$ equals : | [{"identifier": "A", "content": "$$-3$$ "}, {"identifier": "B", "content": "$$-1$$"}, {"identifier": "C", "content": "$$3$$ "}, {"identifier": "D", "content": "$$1$$ "}] | ["A"] | null | $$3x+4y=0$$ is one of the lines of the pair
<br><br>$$6{x^2} - xy + 4c{y^2} = 0,$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
<br><br>Put $$y = - {3 \over 4}x,$$
<br><br>we get $$6{x^2} + {3 \over 4}{x^2} + 4c{\left( { - {3 \over 4}x} \right)^2} = 0$$
<br><br>$$ \Rightarrow 6 + {3 \over 4} + {{9c} \over 4} = 0 \Rightarrow c... | mcq | aieee-2004 |
lCN4wgVHnbo8LVq1 | maths | straight-lines-and-pair-of-straight-lines | pair-of-straight-lines | If one of the lines of $$m{y^2} + \left( {1 - {m^2}} \right)xy - m{x^2} = 0$$ is a bisector of angle between the lines $$xy = 0,$$ then $$m$$ is : | [{"identifier": "A", "content": "$$1$$"}, {"identifier": "B", "content": "$$2$$ "}, {"identifier": "C", "content": "$$-1/2$$ "}, {"identifier": "D", "content": "$$-2$$"}] | ["A"] | null | Equation of bisectors of lines, $$xy=0$$ are $$y = \pm x$$
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264578/exam_images/cudjhzff4ynokmmza2f9.webp" loading="lazy" alt="AIEEE 2007 Mathematics - Straight Lines and Pair of Straight Lines Question 133 English Explanatio... | mcq | aieee-2007 |
1krznn46g | maths | straight-lines-and-pair-of-straight-lines | pair-of-straight-lines | Let the equation of the pair of lines, y = px and y = qx, can be written as (y $$-$$ px) (y $$-$$ qx) = 0. Then the equation of the pair of the angle bisectors of the lines x<sup>2</sup> $$-$$ 4xy $$-$$ 5y<sup>2</sup> = 0 is : | [{"identifier": "A", "content": "x<sup>2</sup> $$-$$ 3xy + y<sup>2</sup> = 0"}, {"identifier": "B", "content": "x<sup>2</sup> + 4xy $$-$$ y<sup>2</sup> = 0"}, {"identifier": "C", "content": "x<sup>2</sup> + 3xy $$-$$ y<sup>2</sup> = 0"}, {"identifier": "D", "content": "x<sup>2</sup> $$-$$ 3xy $$-$$ y<sup>2</sup> = 0"}] | ["C"] | null | Equation of angle bisector of homogeneous <br>equation of pair of straight line ax<sup>2</sup>
+ 2hxy + by<sup>2</sup>
is
<br><br>$${{{x^2} - {y^2}} \over {a - b}} = {{xy} \over h}$$
<br><br>for x<sup>2</sup> – 4xy – 5y<sup>2</sup>
= 0
<br><br> a = 1, h = – 2, b = – 5
<br><br>So, equation of angle bisector is
<br><b... | mcq | jee-main-2021-online-25th-july-evening-shift |
lsana1e6 | maths | straight-lines-and-pair-of-straight-lines | pair-of-straight-lines | The lines $\mathrm{L}_1, \mathrm{~L}_2, \ldots, \mathrm{L}_{20}$ are distinct. For $\mathrm{n}=1,2,3, \ldots, 10$ all the lines $\mathrm{L}_{2 \mathrm{n}-1}$ are parallel to each other and all the lines $L_{2 n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set... | [] | null | 101 | <p>To find the maximum number of points of intersection of pairs of lines from the given set, we need to consider how the lines are arranged based on the given conditions.</p><p>Firstly, there are 10 lines (${L}_1, {L}_3, ..., {L}_{19}$) that are parallel to each other. Since parallel lines do not intersect with each o... | integer | jee-main-2024-online-1st-february-evening-shift |
48fQMDjK28k2Jmox | maths | straight-lines-and-pair-of-straight-lines | position-of-a-point-with-respect-to-a-line | If $$\left( {a,{a^2}} \right)$$ falls inside the angle made by the lines $$y = {x \over 2},$$ $$x > 0$$ and $$y = 3x,$$ $$x > 0,$$ then a belong to : | [{"identifier": "A", "content": "$$\\left( {0,{1 \\over 2}} \\right)$$ "}, {"identifier": "B", "content": "$$\\left( {3,\\infty } \\right)$$ "}, {"identifier": "C", "content": "$$\\left( {{1 \\over 2},3} \\right)$$ "}, {"identifier": "D", "content": "$$\\left( {-3,-{1 \\over 2}} \\right)$$"}] | ["C"] | null | Clearly for point $$P,$$
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266344/exam_images/ctbame8e1bgmyuficejz.webp" loading="lazy" alt="AIEEE 2006 Mathematics - Straight Lines and Pair of Straight Lines Question 136 English Explanation">
<br><br>$${a^2} - 3a < 0$$ ... | mcq | aieee-2006 |
GjQYnrzI4iqIKureOsjgy2xukezf9uo6 | maths | straight-lines-and-pair-of-straight-lines | position-of-a-point-with-respect-to-a-line | The set of all possible values of
$$\theta $$ in the interval
<br/>(0, $$\pi $$) for which the points (1, 2) and (sin
$$\theta $$, cos $$\theta $$) lie <br/>on the same side of the line x + y =
1 is :
| [{"identifier": "A", "content": "$$\\left( {0,{\\pi \\over 4}} \\right)$$"}, {"identifier": "B", "content": "$$\\left( {0,{{3\\pi } \\over 4}} \\right)$$"}, {"identifier": "C", "content": "$$\\left( {{\\pi \\over 4},{{3\\pi } \\over 4}} \\right)$$"}, {"identifier": "D", "content": "$$\\left( {0,{\\pi \\over 2}} \\ri... | ["D"] | null | Let f(x, y) = x + y - 1<br><br>
$$ \because f\left( {1,2} \right).f\left( {\sin \theta ,\cos \theta } \right) > 0$$<br><br>
$$ \Rightarrow 2\left[ {\sin \theta + \cos \theta - 1} \right] > 0$$<br><br>
$$ \Rightarrow \sin \theta + \cos \theta > 1$$<br><br>
$$ \Rightarrow \sin \left( {\theta + {\pi \over 4... | mcq | jee-main-2020-online-2nd-september-evening-slot |
RTESslESMCU0Xz8Ntajgy2xukg3bb6q3 | maths | straight-lines-and-pair-of-straight-lines | position-of-a-point-with-respect-to-a-line | Let L denote the line in the xy-plane with x and
y intercepts as 3 and 1 respectively. Then the
image of the point (–1, –4) in this line is :
| [{"identifier": "A", "content": "$$\\left( {{{11} \\over 5},{{28} \\over 5}} \\right)$$"}, {"identifier": "B", "content": "$$\\left( {{{29} \\over 5},{{11} \\over 5}} \\right)$$"}, {"identifier": "C", "content": "$$\\left( {{{29} \\over 5},{8 \\over 5}} \\right)$$"}, {"identifier": "D", "content": "$$\\left( {{8 \\over... | ["A"] | null | Line is $${x \over 3} + {y \over 1} = 1$$
<br><br>$$ \Rightarrow $$ x + 3y – 3 = 0
<br><br>Let Image of point (–1, –4) is ($$\alpha $$, $$\beta $$)
<br><br>Hence, $${{\alpha + 1} \over 1} = {{\beta + 4} \over 3} = - 2\left( {{{ - 1 - 12 - 3} \over {10}}} \right)$$
<br><br>$$ \Rightarrow $$ $${{\alpha + 1} \over 1} ... | mcq | jee-main-2020-online-6th-september-evening-slot |
xEGrouMGvUDJjnlrut1kls5d2xb | maths | straight-lines-and-pair-of-straight-lines | position-of-a-point-with-respect-to-a-line | The image of the point (3, 5) in the line x $$-$$ y + 1 = 0, lies on : | [{"identifier": "A", "content": "(x $$-$$ 4)<sup>2</sup> + (y $$-$$ 4)<sup>2</sup> = 8"}, {"identifier": "B", "content": "(x $$-$$ 4)<sup>2</sup> + (y $$+$$ 2)<sup>2</sup> = 16"}, {"identifier": "C", "content": "(x $$-$$ 2)<sup>2</sup> + (y $$-$$ 2)<sup>2</sup> = 12"}, {"identifier": "D", "content": "(x $$-$$ 2)<sup>2<... | ["D"] | null | So, let the image is (x, y)<br><br>So, we have<br><br>$${{x - 3} \over 1} = {{y - 5} \over { - 1}} = - {{2(3 - 5 + 1)} \over {1 + 1}}$$<br><br>$$ \Rightarrow $$ x = 4, y = 4<br><br>$$ \Rightarrow $$ Point (4, 4)<br><br>Which will satisfy the curve <br><br>(x $$-$$ 2)<sup>2</sup> + (y $$-$$ 4)<sup>2</sup> = 4<br><br>as... | mcq | jee-main-2021-online-25th-february-morning-slot |
jaoe38c1lscnigca | maths | straight-lines-and-pair-of-straight-lines | position-of-a-point-with-respect-to-a-line | <p>Let $$\mathrm{R}$$ be the interior region between the lines $$3 x-y+1=0$$ and $$x+2 y-5=0$$ containing the origin. The set of all values of $$a$$, for which the points $$\left(a^2, a+1\right)$$ lie in $$R$$, is :</p> | [{"identifier": "A", "content": " $$(-3,0) \\cup\\left(\\frac{2}{3}, 1\\right)$$\n"}, {"identifier": "B", "content": "$$(-3,0) \\cup\\left(\\frac{1}{3}, 1\\right)$$\n"}, {"identifier": "C", "content": "$$(-3,-1) \\cup\\left(\\frac{1}{3}, 1\\right)$$\n"}, {"identifier": "D", "content": "$$(-3,-1) \\cup\\left(-\\fra... | ["B"] | null | <p>$$\begin{aligned}
& \mathrm{P}\left(\mathrm{a}^2, \mathrm{a}+1\right) \\
& \mathrm{L}_1=3 \mathrm{x}-\mathrm{y}+1=0
\end{aligned}$$</p>
<p>Origin and $$\mathrm{P}$$ lies same side w.r.t. $$\mathrm{L}_1$$</p>
<p>$$\begin{aligned}
& \Rightarrow \mathrm{L}_1(0) \cdot \mathrm{L}_1(\mathrm{P})>0 \\
& \... | mcq | jee-main-2024-online-27th-january-evening-shift |
zrvlgkTshq7OXXx6 | maths | straight-lines-and-pair-of-straight-lines | section-formula | The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y-intercept -4. Then a possible value of k is : | [{"identifier": "A", "content": "1 "}, {"identifier": "B", "content": "2 "}, {"identifier": "C", "content": "-2"}, {"identifier": "D", "content": "-4"}] | ["D"] | null | Slope of $$PQ = {{3 - 4} \over {k - 1}} = {{ - 1} \over {k - 1}}$$
<br><br>$$\therefore$$ Slope of perpendicular bisector of
<br><br>$$PQ = \left( {k - 1} \right)$$
<br><br>Also mid point of
<br><br>$$PQ\left( {{{k + 1} \over 2},{7 \over 2}} \right).$$
<br><br>Equation of perpendicular bisector is
<br><br>$$y - {7 \... | mcq | aieee-2008 |
GqfASFWCWxm72l58 | maths | straight-lines-and-pair-of-straight-lines | section-formula | If the line $$2x + y = k$$ passes through the point which divides the line segment joining the points $$(1, 1)$$ and $$(2, 4)$$ in the ratio $$3 : 2$$, then $$k$$ equals : | [{"identifier": "A", "content": "$${{29 \\over 5}}$$"}, {"identifier": "B", "content": "$$5$$"}, {"identifier": "C", "content": "$$6$$ "}, {"identifier": "D", "content": "$${{11 \\over 5}}$$"}] | ["C"] | null | <p>The point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio 3 : 2 is</p>
<p>$$ = \left( {{{3 \times 2 + 2 \times 1} \over {3 + 2}},{{3 \times 4 + 2 \times 1} \over {3 + 2}}} \right)$$</p>
<p>$$ = \left( {{{6 + 2} \over 5},{{12 + 2} \over 5}} \right) = \left( {{8 \over 5},{{14} \over 5}... | mcq | aieee-2012 |
RqP4GjPbQFXkv1uZ | maths | straight-lines-and-pair-of-straight-lines | section-formula | Let $$PS$$ be the median of the triangle with vertices $$P(2, 2)$$, $$Q(6, -1)$$ and $$R(7, 3)$$. The equation of the line passing through $$(1, -1)$$ band parallel to PS is : | [{"identifier": "A", "content": "$$4x + 7y + 3 = 0$$ "}, {"identifier": "B", "content": "$$2x - 9y - 11 = 0$$"}, {"identifier": "C", "content": "$$4x - 7y - 11 = 0$$"}, {"identifier": "D", "content": "$$2x + 9y + 7 = 0$$"}] | ["D"] | null | Let $$P,Q,R,$$ be the vertices of $$\Delta PQR$$
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263525/exam_images/hrqnkqyxxwu0ays0itme.webp" loading="lazy" alt="JEE Main 2014 (Offline) Mathematics - Straight Lines and Pair of Straight Lines Question 124 English Explanat... | mcq | jee-main-2014-offline |
tteS99KoDoR8VPgKcQ97i | maths | straight-lines-and-pair-of-straight-lines | section-formula | A straight line through origin O meets the lines 3y = 10 − 4x and 8x + 6y + 5 = 0 at points A and B respectively. Then O divides the segment AB in the ratio : | [{"identifier": "A", "content": "2 : 3 "}, {"identifier": "B", "content": "1 : 2"}, {"identifier": "C", "content": "4 : 1"}, {"identifier": "D", "content": "3 : 4"}] | ["C"] | null | The lines 4x + 3y $$-$$ 10 = 0 and
<br><br>8x + 6y + 5 = 0 , are parallel as
<br><br> $${4 \over 8}$$ = $${3 \over 6}$$
<br><br>Now length of perpendicular from
<br><br>(0, 0, 0) to 4x + 3y $$-$$ 10 = 0 is,
<br><br>P<sub>1</sub> = $$\left| {{{4\l... | mcq | jee-main-2016-online-10th-april-morning-slot |
YOxO9TpRClzGuqXh6Wjgy2xukf8zo2oh | maths | straight-lines-and-pair-of-straight-lines | section-formula | Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with points A
and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in m)
above the line AC is : | [{"identifier": "A", "content": "10/3"}, {"identifier": "B", "content": "5"}, {"identifier": "C", "content": "20/3"}, {"identifier": "D", "content": "6"}] | ["D"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264412/exam_images/wsbio4fzzawvxpgqqiq9.webp"><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265203/exam_images/ovvgpzy8at9xuqen3rrd.webp" style="max-width: 100%;height: auto;display: block;margi... | mcq | jee-main-2020-online-4th-september-morning-slot |
1l567opn9 | maths | straight-lines-and-pair-of-straight-lines | section-formula | <p>A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2 : 1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle P... | [] | null | 31 | <p> <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5oc3byi/c98d69fd-398a-42cd-a87f-3ee77ee795a4/cca73aa0-0544-11ed-987f-3938cfc0f7f1/file-1l5oc3byj.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5oc3byi/c98d69fd-398a-42cd-a87f-3ee77ee795a4/cca73aa0-0544-11ed-987f-3938cfc0f7f... | integer | jee-main-2022-online-28th-june-morning-shift |
lv5gsk1h | maths | straight-lines-and-pair-of-straight-lines | section-formula | <p>The equations of two sides $$\mathrm{AB}$$ and $$\mathrm{AC}$$ of a triangle $$\mathrm{ABC}$$ are $$4 x+y=14$$ and $$3 x-2 y=5$$, respectively. The point $$\left(2,-\frac{4}{3}\right)$$ divides the third side $$\mathrm{BC}$$ internally in the ratio $$2: 1$$, the equation of the side $$\mathrm{BC}$$ is</p> | [{"identifier": "A", "content": "$$x+6 y+6=0$$\n"}, {"identifier": "B", "content": "$$x-3 y-6=0$$\n"}, {"identifier": "C", "content": "$$x+3 y+2=0$$\n"}, {"identifier": "D", "content": "$$x-6 y-10=0$$"}] | ["C"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lw8pr3ta/992548a6-ea30-4544-b8d1-c7f041deb6d3/854cb2e0-1337-11ef-9f8d-838c388c326d/file-1lw8pr3tb.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lw8pr3ta/992548a6-ea30-4544-b8d1-c7f041deb6d3/854cb2e0-1337-11ef-9f8d-838c388c326d... | mcq | jee-main-2024-online-8th-april-morning-shift |
0QBTBsX9nVPc2AOX | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | If $${x_1},{x_2},{x_3}$$ and $${y_1},{y_2},{y_3}$$ are both in G.P. with the same common ratio, then the points $$\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)$$ and $$\left( {{x_3},{y_3}} \right)$$ : | [{"identifier": "A", "content": "are vertices of a triangle"}, {"identifier": "B", "content": "lie on a straight line "}, {"identifier": "C", "content": "lie on an ellipse "}, {"identifier": "D", "content": "lie on a circle "}] | ["B"] | null | Taking co-ordinates as
<br><br>$$\left( {{x \over r},{y \over r}} \right);\left( {x,y} \right)\,\,\& \,\,\left( {xr,yr} \right)$$
<br><br>Then slope of line joining
<br><br>$$\left( {{x \over r},{y \over r}} \right),\left( {x,y} \right) = {{y\left( {1 - {1 \over r}} \right)} \over {x\left( {1 - {1 \over r}} \right)... | mcq | aieee-2003 |
65zVUjMq9RnyfkmS | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | A square of side a lies above the $$x$$-axis and has one vertex at the origin. The side passing through the origin makes an angle $$\alpha \left( {0 < \alpha < {\pi \over 4}} \right)$$ with the positive direction of x-axis. The equation of its diagonal not passing through the origin is : | [{"identifier": "A", "content": "$$y\\left( {\\cos \\alpha + \\sin \\alpha } \\right) + x\\left( {\\cos \\alpha - \\sin \\alpha } \\right) = a$$ "}, {"identifier": "B", "content": "$$y\\left( {\\cos \\alpha - \\sin \\alpha } \\right) - x\\left( {\\sin \\alpha - \\cos \\alpha } \\right) = a$$ "}, {"identifier": "C",... | ["A"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265012/exam_images/xq65o2ooxotlgbornwg4.webp" loading="lazy" alt="AIEEE 2003 Mathematics - Straight Lines and Pair of Straight Lines Question 148 English Explanation">
<br><br>Co-ordinate of $$A = \left( {a\,\cos \,\alpha ,\,\,a\,\... | mcq | aieee-2003 |
cL9c59Yxiueh22Ud | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | The equation of the straight line passing through the point $$(4, 3)$$ and making intercepts on the co-ordinate axes whose sum is $$-1$$ is : | [{"identifier": "A", "content": "$${x \\over 2} - {y \\over 3} = 1$$ and $${x \\over -2} +{y \\over 1} = 1$$"}, {"identifier": "B", "content": "$${x \\over 2} - {y \\over 3} = -1$$ and $${x \\over -2} +{y \\over 1} = -1$$"}, {"identifier": "C", "content": "$${x \\over 2} + {y \\over 3} = 1$$ and $${x \\over 2} +{y \\ov... | ["A"] | null | Let the required line be $${x \over a} + {y \over b} = 1.......\left( 1 \right)$$
<br><br>then $$a+b=-1$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.........\left( 2 \right)$$
<br><br>$$(1)$$ passes through $$(4,3), $$ $$ \Rightarrow {4 \over a} + {3 \over b} = 1$$
<br><br>$$ \Rightarrow 4b + 3a = ab\,\,................ | mcq | aieee-2004 |
nxWNixRK8ZREBj7D | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | The line parallel to the $$x$$ - axis and passing through the intersection of the lines $$ax + 2by + 3b = 0$$ and $$bx - 2ay - 3a = 0,$$ where $$(a, b)$$ $$ \ne $$ $$(0, 0)$$ is : | [{"identifier": "A", "content": "below the $$x$$ - axis at a distance of $${3 \\over 2}$$ from it "}, {"identifier": "B", "content": "below the $$x$$ - axis at a distance of $${2 \\over 3}$$ from it "}, {"identifier": "C", "content": "above the $$x$$ - axis at a distance of $${3 \\over 2}$$ from it "}, {"identifier":... | ["A"] | null | The line passing through the intersection of lines
<br><br>$$ax + 2by = 3b = 0$$
<br><br>and $$bx - 2ay - 3a = 0$$ is
<br><br>$$ax + 2by + 3b + \lambda \left( {bx - 2ay - 3a} \right) = 0$$
<br><br>$$ \Rightarrow \left( {a + b\lambda } \right)x + \left( {2b - 2a\lambda } \right)y + 3b - 3\lambda a = 0$$
<br><br>As thi... | mcq | aieee-2005 |
Zla6JGrNhMGxYwmk | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | If non zero numbers $$a, b, c$$ are in $$H.P.,$$ then the straight line $${x \over a} + {y \over b} + {1 \over c} = 0$$ always passes through a fixed point. That point is : | [{"identifier": "A", "content": "$$(-1,2)$$ "}, {"identifier": "B", "content": "$$(-1, -2)$$ "}, {"identifier": "C", "content": "$$(1, -2)$$ "}, {"identifier": "D", "content": "$$\\left( {1, - {1 \\over 2}} \\right)$$ "}] | ["C"] | null | $$a,b,c$$ are in $$H.P. \Rightarrow {1 \over a}.{1 \over b},{1 \over c}$$ are in $$A.P.$$
<br><br>$$ \Rightarrow {2 \over b} = {1 \over a} + {1 \over c}$$
<br><br>$$ \Rightarrow {1 \over a} - {2 \over b} + {1 \over c} = 0$$
<br><br>$$\therefore$$ $${x \over a} + {y \over a} + {1 \over c} = 0$$ passes through $$\left( ... | mcq | aieee-2005 |
OAWySqVefvLlw1nd | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | A straight line through the point $$A (3, 4)$$ is such that its intercept between the axes is bisected at $$A$$. Its equation is : | [{"identifier": "A", "content": "$$x + y = 7$$ "}, {"identifier": "B", "content": "$$3x - 4y + 7 = 0$$ "}, {"identifier": "C", "content": "$$4x + 3y = 24$$ "}, {"identifier": "D", "content": "$$3x + 4y = 25$$ "}] | ["C"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266358/exam_images/znwolrv8chfin230lej7.webp" loading="lazy" alt="AIEEE 2006 Mathematics - Straight Lines and Pair of Straight Lines Question 137 English Explanation">
<br><br>As is the mid point of $$PQ,$$ therefore
<br><br>$${{a +... | mcq | aieee-2006 |
jYK1HsUIIC1y5WHG | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | Let $$a, b, c$$ and $$d$$ be non-zero numbers. If the point of intersection of the lines $$4ax + 2ay + c = 0$$ and $$5bx + 2by + d = 0$$ lies in the fourth quadrant and is equidistant from the two axes then : | [{"identifier": "A", "content": "$$3bc - 2ad = 0$$ "}, {"identifier": "B", "content": "$$3bc + 2ad = 0$$"}, {"identifier": "C", "content": "$$2bc - 3ad = 0$$"}, {"identifier": "D", "content": "$$2bc + 3ad = 0$$"}] | ["A"] | null | <p>Since the point of intersection lies on fourth quadrant and equidistant from the two axes,</p>
<p>i.e., let the point be (k, $$-$$k) and this point satisfies the two equations of the given lines.</p>
<p>$$\therefore$$ 4ak $$-$$ 2ak + c = 0 ......... (1)</p>
<p>and 5bk $$-$$ 2bk + d = 0 ..... (2)</p>
<p>From (1) we g... | mcq | jee-main-2014-offline |
bpCTwPT3438MxJvz | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | Two sides of a rhombus are along the lines, $$x - y + 1 = 0$$ and $$7x - y - 5 = 0$$. If its diagonals intersect at $$(-1, -2)$$, then which one of the following is a vertex of this rhombus? | [{"identifier": "A", "content": "$$\\left( {{{ 1} \\over 3}, - {8 \\over 3}} \\right)$$"}, {"identifier": "B", "content": "$$\\left( - {{{ 10} \\over 3}, - {7 \\over 3}} \\right)$$"}, {"identifier": "C", "content": "$$\\left( { - 3, - 9} \\right)$$ "}, {"identifier": "D", "content": "$$\\left( { - 3, - 8} \\right)$$"... | ["A"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264559/exam_images/k1bzxqvdfbjwl4zlissa.webp" loading="lazy" alt="JEE Main 2016 (Offline) Mathematics - Straight Lines and Pair of Straight Lines Question 121 English Explanation">
<br><br>Let other two sides of rhombus are
<br><b... | mcq | jee-main-2016-offline |
LY8gcmgBjIzs6lsaUq1Mb | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | The point (2, 1) is translated parallel to the line L : x− y = 4 by $$2\sqrt 3 $$ units. If the newpoint Q lies in the third quadrant, then the equation of the line passing through Q and perpendicular to L is : | [{"identifier": "A", "content": "x + y = 2 $$-$$ $$\\sqrt 6 $$"}, {"identifier": "B", "content": "x + y = 3 $$-$$ 3$$\\sqrt 6 $$"}, {"identifier": "C", "content": "x + y = 3 $$-$$ 2$$\\sqrt 6 $$"}, {"identifier": "D", "content": "2x + 2y = 1 $$-$$ $$\\sqrt 6 $$"}] | ["C"] | null | x $$-$$ y = 4
<br><br>To find equation of R
<br><br>slope of L = 0 is 1
<br><br>$$ \Rightarrow $$ slope of QR = $$-$$ 1
<br><br>Let QR is y = mx + c
<br><br>y = $$-$$ x + c
<br><br>x + y $$-$$ c = 0
<br><br>distance of QR from (2, 1) is 2$$\sqrt 3 $$
<br><br>2$$\sqrt 3 $$ = $${{\left| {2 + 1 - c} \righ... | mcq | jee-main-2016-online-9th-april-morning-slot |
vT4E9VT8KM2QVtITg2Uw8 | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | A square, of each side 2, lies above the x-axis and has one vertex at the origin. If
one of the sides passing through the origin makes an angle 30<sup>o</sup> with the positive direction of the x-axis, then the sum of the x-coordinates of the vertices of the square is :
| [{"identifier": "A", "content": "$$2\\sqrt 3 - 1$$ "}, {"identifier": "B", "content": "$$2\\sqrt 3 - 2$$"}, {"identifier": "C", "content": "$$\\sqrt 3 - 2$$"}, {"identifier": "D", "content": "$$\\sqrt 3 - 1$$"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265670/exam_images/k7tcjadeqglnae8jaifx.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2017 (Online) 9th April Morning Slot Mathematics - Straight Lines and Pair of Straight Lines Question 110 Engl... | mcq | jee-main-2017-online-9th-april-morning-slot |
vqNPfS7B1xL53D3pxu4Up | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | The sides of a rhombus ABCD are parallel to the lines, x $$-$$ y + 2 = 0 and 7x $$-$$ y + 3 = 0. If the diagonals of the rhombus intersect P(1, 2) and the vertex A (different from the origin) is on the y-axis, then the coordinate of A is : | [{"identifier": "A", "content": "$${5 \\over 2}$$"}, {"identifier": "B", "content": "$${7 \\over 4}$$"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "$${7 \\over 2}$$"}] | ["A"] | null | Let the coordinate A be (0, c)
<br><br>Equations of the given lines are
<br><br>x $$-$$ y + 2 = 0 and 7x $$-$$ y + 3 = 0
<br><br>We know that the diagonals of the rhombus will be parallel to the angle bisectors of the two given lines; y = x + 2 and y = 7x + 3
<br><br>$$\therefore\,\,\,$$ equation of angle bisectors is... | mcq | jee-main-2018-online-15th-april-evening-slot |
xC0LBb5tjEdVjHbTBY3rsa0w2w9jxb12n1a | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and
the perpendicular from the origin to this line makes an angle of 60<sup>o</sup> with the line x + y = 0. Then an equation
of the line L is : | [{"identifier": "A", "content": "x + $$\\sqrt 3 $$y = 8"}, {"identifier": "B", "content": "$$\\sqrt 3 $$x + y = 8"}, {"identifier": "C", "content": "( $$\\sqrt 3 $$ + 1)x + ( $$\\sqrt 3 $$ \u2013 1)y = 8 $$\\sqrt 2 $$"}, {"identifier": "D", "content": "( $$\\sqrt 3 $$ - 1)x + ( $$\\sqrt 3 $$ + 1)y = 8 $$\\sqrt 2 $... | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263835/exam_images/thzbx3newaxtuygg2q1m.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 12th April Evening Slot Mathematics - Straight Lines and Pair of Straight Lines Question 87 Engl... | mcq | jee-main-2019-online-12th-april-evening-slot |
lHjdEqZBq7U1siUTH43rsa0w2w9jwxcnqce | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | The region represented by| x – y | $$ \le $$ 2 and | x + y| $$ \le $$ 2 is bounded by a : | [{"identifier": "A", "content": "rhombus of area 8$$\\sqrt 2 $$ sq. units"}, {"identifier": "B", "content": "square of side length 2$$\\sqrt 2 $$ units"}, {"identifier": "C", "content": "square of area 16 sq. units"}, {"identifier": "D", "content": "rhombus of side length 2 units "}] | ["B"] | null | $${C_1}{\rm{ }}:{\rm{ }}\left| {y{\rm{ }}-{\rm{ }}x} \right|{\rm{ }} \le {\rm{ }}2$$<br><br>
$${C_2}{\rm{ }}:{\rm{ }}\left| {y{\rm{ + }}x} \right|{\rm{ }} \le {\rm{ }}2$$<br><br>
Now region is square<br><br>
<picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266... | mcq | jee-main-2019-online-10th-april-morning-slot |
11zVmn3h8n66ftAlElPXW | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | Slope of a line passing through P(2, 3) and
intersecting the line, x + y = 7 at a distance of
4 units from P, is : | [{"identifier": "A", "content": "$${{\\sqrt 7 - 1} \\over {\\sqrt 7 + 1}}$$"}, {"identifier": "B", "content": "$${{\\sqrt 5 - 1} \\over {\\sqrt 5 + 1}}$$"}, {"identifier": "C", "content": "$${{1 - \\sqrt 5 } \\over {1 + \\sqrt 5 }}$$"}, {"identifier": "D", "content": "$${{1 - \\sqrt 7 } \\over {1 + \\sqrt 7 }}$$"}] | ["D"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266822/exam_images/vodxfa1zmymjb9pd25x3.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263362/exam_images/jeb4k12c2xkbyeykmyvx.webp"><img src="https://res.c... | mcq | jee-main-2019-online-9th-april-morning-slot |
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