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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 | 8,421 |
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>$$ \Rig... | mcq | jee-main-2017-offline | 8,422 |
oKvo7JHtrCRqrm0obaHah | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | Let the equations of two sides of a triangle be 3x $$-$$ 2y + 6 = 0 and 4x + 5y $$-$$ 20 = 0. If the orthocentre of this triangle is at (1, 1), then the equation of its third side is : | [{"identifier": "A", "content": "122y $$-$$ 26x $$-$$ 1675 = 0"}, {"identifier": "B", "content": "122y + 26x + 1675 = 0"}, {"identifier": "C", "content": "26x + 61y + 1675 = 0"}, {"identifier": "D", "content": "26x $$-$$ 122y $$-$$ 1675 = 0"}] | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264905/exam_images/ubp63holkjbubzmm73xt.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 9th January Evening Slot Mathematics - Straight Lines and Pair of Straight Lines Question 104 En... | mcq | jee-main-2019-online-9th-january-evening-slot | 8,423 |
jzTjddlv24Pe3XwUGoOyx | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | A point P moves on the line 2x – 3y + 4 = 0. If Q(1, 4) and R (3, – 2) are fixed points, then the locus of the centroid of $$\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)$$
<br><br>Poin... | mcq | jee-main-2019-online-10th-january-morning-slot | 8,424 |
tkJLUql0mThR9SD5j5Xie | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | If the line 3x + 4y – 24 = 0 intersects the x-axis at the point A and the y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is : | [{"identifier": "A", "content": "(3, 4)"}, {"identifier": "B", "content": "(2, 2)"}, {"identifier": "C", "content": "(4, 4)"}, {"identifier": "D", "content": "(4, 3)"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265477/exam_images/migtvndabwdpqcwl9h0o.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 10th January Morning Slot Mathematics - Straight Lines and Pair of Straight Lines Question 101 E... | mcq | jee-main-2019-online-10th-january-morning-slot | 8,425 |
TWc5X8nsXEQEf4RxhueyF | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | Two vertices of a triangle are (0, 2) and (4, 3). If its orthocenter is at the origin, then its third vertex lies in which quadrant : | [{"identifier": "A", "content": "third"}, {"identifier": "B", "content": "fourth"}, {"identifier": "C", "content": "second"}, {"identifier": "D", "content": "first"}] | ["C"] | null | m<sub>BD</sub> $$ \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 $$ b + 4a = 0 . . . . (i)
<br><br><img src="https://res.cloudinary... | mcq | jee-main-2019-online-10th-january-evening-slot | 8,426 |
kAbgo0MsBNIlywerYv7k9k2k5itigqx | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | Let C be the centroid of the triangle with
vertices (3, –1), (1, 3) and (2, 4). Let P be the
point of intersection of the lines x + 3y – 1 = 0
and 3x – y + 1 = 0. Then the line passing through
the points C and P also passes through the
point : | [{"identifier": "A", "content": "(\u20139, \u20137)"}, {"identifier": "B", "content": "(9, 7)"}, {"identifier": "C", "content": "(7, 6)"}, {"identifier": "D", "content": "(\u20139, \u20136)"}] | ["D"] | null | Centroid C $$\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)$$
<br><br>So, equation of line CP is 8x – 11y + 6 = 0
<br><br>Point (–9, –6) satisfy this equation. | mcq | jee-main-2020-online-9th-january-morning-slot | 8,428 |
2l8AfZuROD0j15pyvo1kmja51tj | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | In a triangle PQR, the co-ordinates of the points P and Q are ($$-$$2, 4) and (4, $$-$$2) respectively. If the equation of the perpendicular bisector of PR is 2x $$-$$ y + 2 = 0, then the centre of the circumcircle of the $$\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$$<br><br>So... | mcq | jee-main-2021-online-17th-march-morning-shift | 8,430 |
0EJF9pQuqeuwViMpNb1kmknr7sp | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | Let tan$$\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$$ + cos<sup>3</sup> $... | integer | jee-main-2021-online-17th-march-evening-shift | 8,431 |
Qjcr2pxjkzDbBSuAFH1kmm3hfre | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x + y = 3. If R and r be the radius of circumcircle and incircle respectively of $$\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 $$"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264797/exam_images/ajzrnczcpsrafxwpsdbk.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 18th March Evening Shift Mathematics - Straight Lines and Pair of Straight Lines Question 65 Engli... | mcq | jee-main-2021-online-18th-march-evening-shift | 8,432 |
1krrwe4qx | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | Consider a triangle having vertices A($$-$$2, 3), B(1, 9) and C(3, 8). If a line L passing through the circum-centre of triangle ABC, bisects line BC, and intersects y-axis at point $$\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 | 8,433 |
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 | 8,434 |
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 | 8,435 |
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)$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7qa6xtq/139436a6-31d3-4607-a05c-66e2b0bc9d4d/5ec55de0-2def-11ed-a744-1fb8f3709cfa/file-1l7qa6xtr.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7qa6xtq/139... | mcq | jee-main-2022-online-27th-july-evening-shift | 8,437 |
1l6p2cpyz | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1,1). If the line AP intersects the line BC at the point Q$$\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}$$"}, {"identifier": "D", "content": "4"}] | ["B"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7ssjo3c/b55f47a7-9a89-4d60-875b-d115e046f61f/b6651380-2f50-11ed-85dd-19dc023e9ad1/file-1l7ssjo3d.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7ssjo3c/b55f47a7-9a89-4d60-875b-d115e046f61f/b6651380-2f50-11ed-85dd-19dc023e9ad1... | mcq | jee-main-2022-online-29th-july-morning-shift | 8,438 |
1ldonjeqx | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>If the orthocentre of the triangle, whose vertices are (1, 2), (2, 3) and (3, 1) is $$(\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$$"}] | ["A"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lf41l718/e00587c4-d3cd-4c76-ac3c-9d17149d6abd/35cb41c0-c016-11ed-89c7-f9c9b186ec3c/file-1lf41l719.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lf41l718/e00587c4-d3cd-4c76-ac3c-9d17149d6abd/35cb41c0-c016-11ed-89c7-f9c9b186ec3c/fi... | mcq | jee-main-2023-online-1st-february-morning-shift | 8,440 |
lgny7des | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | If $(\alpha, \beta)$ is the orthocenter of the triangle $\mathrm{ABC}$ with vertices $A(3,-7), B(-1,2)$ and $C(4,5)$, then $9 \alpha-6 \beta+60$ is equal to : | [{"identifier": "A", "content": "30"}, {"identifier": "B", "content": "40"}, {"identifier": "C", "content": "25"}, {"identifier": "D", "content": "35"}] | ["C"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lgxuayzx/c68e4c1e-f2ed-4189-96d3-77fa5b78bc40/7473b5d0-e445-11ed-97cc-4f9b4bf32610/file-1lgxuayzy.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lgxuayzx/c68e4c1e-f2ed-4189-96d3-77fa5b78bc40/7473b5d0-e445-11ed-97cc-4f9b4bf32610/fi... | mcq | jee-main-2023-online-15th-april-morning-shift | 8,442 |
1lgowjxyl | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>Let $$(\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$$"}] | ["B"] | null | <ol>
<li>Solve the equations $15x - y = 82$ and $6x - 5y = -4$</li>
</ol>
<p>Multiply the first equation by 5 and the second by 1 and then subtract the second from the first:</p>
<p>$75x - 5y = 410$</p>
<p>$6x - 5y = -4$</p>
<p>Subtracting these gives $69x = 414$ which leads to $x = 6$</p>
<p>Substitute $x = 6$ into th... | mcq | jee-main-2023-online-13th-april-evening-shift | 8,443 |
lv5gs2w8 | maths | straight-lines-and-pair-of-straight-lines | centers-of-triangle | <p>If the orthocentre of the triangle formed by the lines $$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$$ respectively</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lw8vj8jo/f80a9888-77a1-4003-a6a8-8ff4c790fc36/21421350-134e-11ef-8bbe-1b4949638519/file-1lw8vj8jp.png?format=png" data-orsrc="https://ap... | integer | jee-main-2024-online-8th-april-morning-shift | 8,445 |
1ktocacrw | maths | straight-lines-and-pair-of-straight-lines | distance-formula | A man starts walking from the point P($$-$$3, 4), touches the x-axis at R, and then turns to reach at the point Q(0, 2). The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then $$50\left( {{{(PR)}^2} + {{(RQ)}^2}} \right)$$ is equal to ____________. | [] | null | 1250 | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1kwjjmzcw/b86a6d68-7b5e-434f-a89c-eef8431d96b0/715950f0-5074-11ec-8a49-997c456a58f9/file-1kwjjmzcx.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1kwjjmzcw/b86a6d68-7b5e-434f-a89c-eef8431d96b0/715950f0-5074-11ec-8a49-997c456a58f9/fi... | integer | jee-main-2021-online-1st-september-evening-shift | 8,447 |
1l545buee | maths | straight-lines-and-pair-of-straight-lines | distance-formula | <p>The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A' B (where B is the point (2, 3)) subtend angle $${\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 | 8,448 |
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>$$\,\,\,\,\,\,\,\,D = {{{a^2} - a + 1} \over {\sqrt 2 }} = {1 \over {\sqrt 2 }}\left[ {{{\left( {a - {1 \over 2}} \right)}... | mcq | aieee-2009 | 8,449 |
EXLH2cD628vv64gu | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | The line $$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 | 8,450 |
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 = $${{ - \left( {\left( {3 \times 0} \right) + \left( {1 \times... | mcq | jee-main-2018-online-15th-april-evening-slot | 8,451 |
juSWMIEgIH0JUKssdD3rsa0w2w9jx2ezrxa | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | Lines are drawn parallel to the line 4x – 3y + 2 = 0, at a distance
$${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 | 8,452 |
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>$$6x - 3y + 9 = 0... | integer | jee-main-2020-online-5th-september-morning-slot | 8,453 |
1ktiq1mtm | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | If p and q are the lengths of the perpendiculars from the origin on the lines,<br/><br/>x cosec $$\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|$$ .... (i... | mcq | jee-main-2021-online-31st-august-morning-shift | 8,454 |
1l6f18gvo | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | <p>Let the point $$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)$$ is equal to :</p> | [{"identifier": "A", "content": "$$-$$14"}, {"identifier": "B", "content": "42"}, {"identifier": "C", "content": "$$-$$22"}, {"identifier": "D", "content": "14"}] | ["D"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7bukhhd/39a595d1-0f44-4de9-a640-89709cea02f6/4c50df10-25ff-11ed-9c74-c5a04899a045/file-1l7bukhhe.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7bukhhd/39a595d1-0f44-4de9-a640-89709cea02f6/4c50df10-25ff-11ed-9c74-c5a04899a045... | mcq | jee-main-2022-online-25th-july-evening-shift | 8,455 |
jaoe38c1lscoky97 | maths | straight-lines-and-pair-of-straight-lines | distance-of-a-point-from-a-line | <p>If the sum of squares of all real values of $$\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 | 8,457 |
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 | 8,458 |
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 | 8,459 |
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 | 8,462 |
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 = $${3 \over ... | mcq | jee-main-2019-online-9th-january-morning-slot | 8,463 |
1MTF5OuuQpiWHnCU | maths | straight-lines-and-pair-of-straight-lines | locus | Locus of mid point of the portion between the axes of
<br/><br/>$$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>$$x\cos \alpha + y\sin \... | mcq | aieee-2002 | 8,464 |
2Jxdn4DG7BLAO7DQ | maths | straight-lines-and-pair-of-straight-lines | locus | Locus of centroid of the triangle whose vertices are $$\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 | 8,465 |
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 | 8,466 |
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>$$ \Rightarro... | mcq | aieee-2004 | 8,467 |
3BmWgWmgNz7RCJsg8h9jZ | maths | straight-lines-and-pair-of-straight-lines | locus | Let O(0, 0) and A(0, 1) be two fixed points. Then
the locus of a point P such that the perimeter of
$$\Delta $$AOP is 4, is : | [{"identifier": "A", "content": "9x<sup>2</sup> + 8y<sup>2</sup> \u2013 8y = 16\n"}, {"identifier": "B", "content": "8x<sup>2</sup> \u2013 9y<sup>2</sup> + 9y = 18"}, {"identifier": "C", "content": "8x<sup>2</sup> + 9y<sup>2</sup> \u2013 9y = 18\n"}, {"identifier": "D", "content": "9x<sup>2</sup> \u2013 8y<sup>2</sup> ... | ["A"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267479/exam_images/wjrxugubrcihnruos7c4.webp"><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265282/exam_images/uonftnxl941nxalen4tg.webp" style="max-width: 100%;height: auto;display: block;margi... | mcq | jee-main-2019-online-8th-april-morning-slot | 8,470 |
fsvYqfrKekcQWsXuJU1kmlm4kfs | maths | straight-lines-and-pair-of-straight-lines | locus | A square ABCD has all its vertices on the curve x<sup>2</sup>y<sup>2</sup> = 1. The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is _________. | [] | null | 80 | x<sup>2</sup>y<sup>2</sup> = 1
<br><br>$$ \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 | 8,472 |
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 | 8,473 |
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 | 8,474 |
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 | 8,475 |
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 | 8,477 |
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 | 8,479 |
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 $$(2)$$ $${q ... | mcq | aieee-2003 | 8,480 |
LKV0RhPLyjQeTiXA | maths | straight-lines-and-pair-of-straight-lines | pair-of-straight-lines | If the sum of the slopes of the lines given by $${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$$ | mcq | aieee-2004 | 8,481 |
Cy25MQi1BYWBv2tA | maths | straight-lines-and-pair-of-straight-lines | pair-of-straight-lines | If one of the lines given by $$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 | 8,482 |
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 | 8,483 |
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 | 8,484 |
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 | 8,485 |
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 | 8,487 |
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 | 8,488 |
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 | 8,489 |
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>$$y - {7 \... | mcq | aieee-2008 | 8,491 |
GqfASFWCWxm72l58 | maths | straight-lines-and-pair-of-straight-lines | section-formula | If the line $$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 | 8,492 |
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 | 8,493 |
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> = $$\left| {{{4\l... | mcq | jee-main-2016-online-10th-april-morning-slot | 8,494 |
1l567opn9 | maths | straight-lines-and-pair-of-straight-lines | section-formula | <p>A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2 : 1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle P... | [] | null | 31 | <p> <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5oc3byi/c98d69fd-398a-42cd-a87f-3ee77ee795a4/cca73aa0-0544-11ed-987f-3938cfc0f7f1/file-1l5oc3byj.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5oc3byi/c98d69fd-398a-42cd-a87f-3ee77ee795a4/cca73aa0-0544-11ed-987f-3938cfc0f7f... | integer | jee-main-2022-online-28th-june-morning-shift | 8,496 |
lv5gsk1h | maths | straight-lines-and-pair-of-straight-lines | section-formula | <p>The equations of two sides $$\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$$"}] | ["C"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lw8pr3ta/992548a6-ea30-4544-b8d1-c7f041deb6d3/854cb2e0-1337-11ef-9f8d-838c388c326d/file-1lw8pr3tb.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lw8pr3ta/992548a6-ea30-4544-b8d1-c7f041deb6d3/854cb2e0-1337-11ef-9f8d-838c388c326d... | mcq | jee-main-2024-online-8th-april-morning-shift | 8,497 |
0QBTBsX9nVPc2AOX | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | If $${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 | 8,498 |
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 | 8,500 |
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 | 8,501 |
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 $$\left( ... | mcq | aieee-2005 | 8,502 |
OAWySqVefvLlw1nd | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | A straight line through the point $$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 | 8,503 |
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$$\sqrt 3 $$ = $${{\left| {2 + 1 - c} \righ... | mcq | jee-main-2016-online-9th-april-morning-slot | 8,506 |
vT4E9VT8KM2QVtITg2Uw8 | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | A square, of each side 2, lies above the x-axis and has one vertex at the origin. If
one of the sides passing through the origin makes an angle 30<sup>o</sup> with the positive direction of the x-axis, then the sum of the x-coordinates of the vertices of the square is :
| [{"identifier": "A", "content": "$$2\\sqrt 3 - 1$$ "}, {"identifier": "B", "content": "$$2\\sqrt 3 - 2$$"}, {"identifier": "C", "content": "$$\\sqrt 3 - 2$$"}, {"identifier": "D", "content": "$$\\sqrt 3 - 1$$"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265670/exam_images/k7tcjadeqglnae8jaifx.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2017 (Online) 9th April Morning Slot Mathematics - Straight Lines and Pair of Straight Lines Question 110 Engl... | mcq | jee-main-2017-online-9th-april-morning-slot | 8,507 |
vqNPfS7B1xL53D3pxu4Up | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | The sides of a rhombus ABCD are parallel to the lines, x $$-$$ y + 2 = 0 and 7x $$-$$ y + 3 = 0. If the diagonals of the rhombus intersect P(1, 2) and the vertex A (different from the origin) is on the y-axis, then the coordinate of A is : | [{"identifier": "A", "content": "$${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\,\,\,$$ equation of angle bisectors is... | mcq | jee-main-2018-online-15th-april-evening-slot | 8,508 |
xC0LBb5tjEdVjHbTBY3rsa0w2w9jxb12n1a | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and
the perpendicular from the origin to this line makes an angle of 60<sup>o</sup> with the line x + y = 0. Then an equation
of the line L is : | [{"identifier": "A", "content": "x + $$\\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 | 8,509 |
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 | 8,510 |
ssSsGJs4IgV0TRS9sbJkO | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | If the system of linear equations<br/><br/>
x – 2y + kz = 1<br/>
2x + y + z = 2<br/>
3x – y – kz = 3<br/><br/>
has a solution (x,y,z), z $$ \ne $$ 0, then (x,y) lies on
the straight line whose equation is : | [{"identifier": "A", "content": "4x \u2013 3y \u2013 4 = 0"}, {"identifier": "B", "content": "3x \u2013 4y \u2013 1 = 0"}, {"identifier": "C", "content": "4x \u2013 3y \u2013 1 = 0"}, {"identifier": "D", "content": "3x \u2013 4y \u2013 4 = 0"}] | ["A"] | null | x – 2y + kz = 1 ......(1)<br><br>
2x + y + z = 2 .........(2)<br><br>
3x – y – kz = 3 ........(3)
<br><br>for locus of (x, y)
add equation (1) + (3)
<br><br>4x – 3y = 4
<br><br>$$ \Rightarrow $$ 4x – 3y - 4 = 0 | mcq | jee-main-2019-online-8th-april-evening-slot | 8,512 |
OoONgLpzJPuR08YqU4ZzC | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | If a straight line passing through the point P(–3, 4) is such that its intercepted portion between the coordinate axes is bisected at P, then its equation is : | [{"identifier": "A", "content": "x \u2013 y + 7 = 0"}, {"identifier": "B", "content": "4x \u2013 3y + 24 = 0"}, {"identifier": "C", "content": "4x + 3y = 0"}, {"identifier": "D", "content": "3x \u2013 4y + 25 = 0"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263333/exam_images/sgkffsp4mfuv6idliumw.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 12th January Evening Slot Mathematics - Straight Lines and Pair of Straight Lines Question 96 En... | mcq | jee-main-2019-online-12th-january-evening-slot | 8,513 |
DpbIaKmUJaLDxujcxegRK | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3, 4) and (2, 5), then the
equation of the diagonal AD is : | [{"identifier": "A", "content": "5x + 3y \u2013 11 = 0"}, {"identifier": "B", "content": "5x \u2013 3y + 1 = 0"}, {"identifier": "C", "content": "3x \u2013 5y + 7 = 0"}, {"identifier": "D", "content": "3x + 5y \u2013 13 = 0"}] | ["B"] | null | co-ordinates of point D are (4, 7)
<br><br>$$ \Rightarrow $$ line AD is 5x $$-$$ 3y + 1 = 0 | mcq | jee-main-2019-online-11th-january-evening-slot | 8,514 |
LMDd8M7tN0EYa27fmeMWM | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | Two sides of a parallelogram are along the lines, x + y = 3 & x – y + 3 = 0. If its diagonals intersect at (2, 4), then one of its vertex is : | [{"identifier": "A", "content": "(2, 1)"}, {"identifier": "B", "content": "(2, 6)"}, {"identifier": "C", "content": "(3, 5)"}, {"identifier": "D", "content": "(3, 6)"}] | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263689/exam_images/shi6jmtimy8lhgilqtoi.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 10th January Evening Slot Mathematics - Straight Lines and Pair of Straight Lines Question 100 E... | mcq | jee-main-2019-online-10th-january-evening-slot | 8,515 |
kLBkRZEqREDdFmqosAann | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | A point on the straight line, 3x + 5y = 15 which is equidistant from the coordinate axes will lie only
in : | [{"identifier": "A", "content": "1<sup>st </sup> and 2<sup>nd</sup> qudratants"}, {"identifier": "B", "content": "4<sup>th</sup> qudratant"}, {"identifier": "C", "content": "1<sup>st </sup> and 2<sup>nd</sup> and 4<sup>th</sup> qudratants"}, {"identifier": "D", "content": "1<sup>st </sup> qudratant"}] | ["A"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264028/exam_images/c5iqjp4havmvitlpvih5.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264572/exam_images/kiwz4enqututlxqicwwd.webp"><source media="(max-wid... | mcq | jee-main-2019-online-8th-april-morning-slot | 8,516 |
QxqEuxDfNrfjec79WCjgy2xukfahc33t | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | If the perpendicular bisector of the line segment joining the points P(1 ,4) and Q(k, 3) has y-intercept equal to –4, then a value of k is : | [{"identifier": "A", "content": "$$\\sqrt {14} $$"}, {"identifier": "B", "content": "-4"}, {"identifier": "C", "content": "\u20132 "}, {"identifier": "D", "content": "$$\\sqrt {15} $$"}] | ["B"] | null | $${m_{PQ}} = {{4 - 3} \over {1 - k}} $$
<br><br>$$ \therefore $$ Slope of perpendicular bisector of PQ, $$ {m_ \bot } = k - 1$$<br><br>mid point of PQ = $$\left( {{{k + 1} \over 2},{7 \over 2}} \right)$$<br><br>equation of perpendicular bisector<br><br>$$y - {7 \over 2} = (k - 1)\left( {x - {{k + 1} \over 2}} \right)$$... | mcq | jee-main-2020-online-4th-september-evening-slot | 8,517 |
XUXGa7rHrUGBomtNuF1klrg4w6b | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | A man is walking on a straight line. The arithmetic mean
of the reciprocals of the intercepts of this line on the
coordinate axes is $${1 \over 4}$$. Three stones A, B and C are placed at the points
(1, 1), (2, 2) and (4, 4) respectively. Then, which of these stones is / are on the path of the man? | [{"identifier": "A", "content": "A only"}, {"identifier": "B", "content": "All the three"}, {"identifier": "C", "content": "C only"}, {"identifier": "D", "content": "B only"}] | ["D"] | null | Given, position of A = (1, 1)<br><br>Position of B = (2, 2)<br><br>Position of C = (4, 4)<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1kxoe9q9z/76596a6c-2785-45d0-8cbb-72347daf83fe/95cbec80-66eb-11ec-b4c9-97a7fc3f3aad/file-1kxoe9qa0.png?format=png" data-orsrc="https://app-content.cdn.examgoal... | mcq | jee-main-2021-online-24th-february-morning-slot | 8,519 |
9CTLjZBJ0UzByUARrD1kmjbnx0l | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | The maximum value of z in the following equation z = 6xy + y<sup>2</sup>, where 3x + 4y $$ \le $$ 100 and 4x + 3y $$ \le $$ 75 for x $$ \ge $$ 0 and y $$ \ge $$ 0 is __________. | [] | null | 904 | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264667/exam_images/acnl84vwiakold3way0b.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 17th March Morning Shift Mathematics - Straight Lines and Pair of Straight Lines Question 70 Engli... | integer | jee-main-2021-online-17th-march-morning-shift | 8,521 |
dmYotEs1QcgiGgtCOh1kmlhxh16 | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is : | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "0"}] | ["B"] | null | 3x + 4(mx + 1) = 9<br><br>$$ \Rightarrow $$ x(3 + 4m) = 5<br><br>$$ \Rightarrow $$ $$x = {5 \over {(3 + 4m)}}$$<br><br>$$ \Rightarrow $$ (3 + 4m) = $$\pm$$1, $$\pm$$5<br><br>$$ \Rightarrow $$ 4m = $$-$$3 $$\pm$$ 1, $$-$$3 $$\pm$$ 5 <br><br>$$ \Rightarrow $$ 4m = $$-$$4, $$-$$2, $$-$$8, 2<br><br>$$ \Rightarrow $$ m = $$... | mcq | jee-main-2021-online-18th-march-morning-shift | 8,522 |
1krxgdh61 | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | The point P (a, b) undergoes the following three transformations successively :<br/><br/>(a) reflection about the line y = x.<br/><br/>(b) translation through 2 units along the positive direction of x-axis.<br/><br/>(c) rotation through angle $${\pi \over 4}$$ about the origin in the anti-clockwise direction.<br/><br/... | [{"identifier": "A", "content": "13"}, {"identifier": "B", "content": "9"}, {"identifier": "C", "content": "5"}, {"identifier": "D", "content": "7"}] | ["B"] | null | Image of A(a, b) along y = x is B(b, a). Translating it 2 units it becomes C(b + 2, a).<br><br>Now, applying rotation theorem<br><br>$$ - {1 \over {\sqrt 2 }} + {7 \over {\sqrt 2 }}i = \left( {(b + 2) + ai} \right)\left( {\cos {\pi \over 4} + i\sin {\pi \over 4}} \right)$$<br><br>$$ - {1 \over {\sqrt 2 }} + {7 \over ... | mcq | jee-main-2021-online-27th-july-evening-shift | 8,523 |
1krxkmgz3 | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0. If the equation of one of the diagonals of the parallelogram is 11x + 7y = 9, then other diagonal passes through the point : | [{"identifier": "A", "content": "(1, 2)"}, {"identifier": "B", "content": "(2, 2)"}, {"identifier": "C", "content": "(2, 1)"}, {"identifier": "D", "content": "(1, 3)"}] | ["B"] | null | Both the lines pass through origin.<br><br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266102/exam_images/h1fc0mfvmn1izevgxbxm.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 27th July Evening Shift Mathematics - Straight Lines an... | mcq | jee-main-2021-online-27th-july-evening-shift | 8,524 |
1l6dw9dqv | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | <p>A line, with the slope greater than one, passes through the point $$A(4,3)$$ and intersects the line $$x-y-2=0$$ at the point B. If the length of the line segment $$A B$$ is $$\frac{\sqrt{29}}{3}$$, then $$B$$ also lies on the line :
</p> | [{"identifier": "A", "content": "$$2 x+y=9$$"}, {"identifier": "B", "content": "$$3 x-2 y=7$$"}, {"identifier": "C", "content": "$$ x+2 y=6$$"}, {"identifier": "D", "content": "$$2 x-3 y=3$$"}] | ["C"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l97rw5km/2f99cd4f-8200-402b-aaa3-f8d0d73a633d/21992d60-4b5a-11ed-bfde-e1cb3fafe700/file-1l97rw5kn.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l97rw5km/2f99cd4f-8200-402b-aaa3-f8d0d73a633d/21992d60-4b5a-11ed-bfde-e1cb3fafe700/fi... | mcq | jee-main-2022-online-25th-july-morning-shift | 8,525 |
1l6rehhn1 | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | <p>Let $$m_{1}, m_{2}$$ be the slopes of two adjacent sides of a square of side a such that $$a^{2}+11 a+3\left(m_{1}^{2}+m_{2}^{2}\right)=220$$. If one vertex of the square is $$(10(\cos \alpha-\sin \alpha), 10(\sin \alpha+\cos \alpha))$$, where $$\alpha \in\left(0, \frac{\pi}{2}\right)$$ and the equation of one diago... | [{"identifier": "A", "content": "119"}, {"identifier": "B", "content": "128"}, {"identifier": "C", "content": "145"}, {"identifier": "D", "content": "155"}] | ["B"] | null | One vertex of square is
<br/><br/>
$(10(\cos \alpha-\sin \alpha), 10(\sin \alpha+\cos \alpha))$
<br/><br/>
and one of the diagonal is
<br/><br/>
$(\cos \alpha-\sin \alpha) x+(\sin \alpha+\cos \alpha) y=10$
<br/><br/>
So the other diagonal can be obtained as
<br/><br/>
$(\cos \alpha+\sin \alpha) x-(\cos \alpha-\sin \alp... | mcq | jee-main-2022-online-29th-july-evening-shift | 8,526 |
1ldsuqxe4 | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | <p>A light ray emits from the origin making an angle 30$$^\circ$$ with the positive $$x$$-axis. After getting reflected by the line $$x+y=1$$, if this ray intersects $$x$$-axis at Q, then the abscissa of Q is :</p> | [{"identifier": "A", "content": "$${2 \\over {\\left( {\\sqrt 3 - 1} \\right)}}$$"}, {"identifier": "B", "content": "$${2 \\over {3 - \\sqrt 3 }}$$"}, {"identifier": "C", "content": "$${{\\sqrt 3 } \\over {2\\left( {\\sqrt 3 + 1} \\right)}}$$"}, {"identifier": "D", "content": "$${2 \\over {3 + \\sqrt 3 }}$$"}] | ["D"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1ldtzo4m3/b85082b3-e539-44fe-bf5f-e63a3546f4f7/a7383ba0-a6c2-11ed-8d92-0101c7cb78f8/file-1ldtzo4m4.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1ldtzo4m3/b85082b3-e539-44fe-bf5f-e63a3546f4f7/a7383ba0-a6c2-11ed-8d92-0101c7cb78f8... | mcq | jee-main-2023-online-29th-january-morning-shift | 8,527 |
1lh20btfj | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | <p>The straight lines $$\mathrm{l_{1}}$$ and $$\mathrm{l_{2}}$$ pass through the origin and trisect the line segment of the line L : $$9 x+5 y=45$$ between the axes. If $$\mathrm{m}_{1}$$ and $$\mathrm{m}_{2}$$ are the slopes of the lines $$\mathrm{l_{1}}$$ and $$\mathrm{l_{2}}$$, then the point of intersection of the ... | [{"identifier": "A", "content": "$$6 x-y=15$$"}, {"identifier": "B", "content": "$$6 x+y=10$$"}, {"identifier": "C", "content": "$$\\mathrm{y}-x=5$$"}, {"identifier": "D", "content": "$$y-2 x=5$$"}] | ["C"] | null | Given line $L: 9 x+5 y=45$ ..........(i)
<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lnmwb3jr/502d615e-8cf5-4478-9d73-0b9837e70b02/2ebbe560-68d6-11ee-a3a1-07c5c60fca90/file-6y3zli1lnmwb3js.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lnmwb3jr/502d615e-... | mcq | jee-main-2023-online-6th-april-morning-shift | 8,528 |
jaoe38c1lsd505jf | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | <p>Let $$A(-2,-1), B(1,0), C(\alpha, \beta)$$ and $$D(\gamma, \delta)$$ be the vertices of a parallelogram $$A B C D$$. If the point $$C$$ lies on $$2 x-y=5$$ and the point $$D$$ lies on $$3 x-2 y=6$$, then the value of $$|\alpha+\beta+\gamma+\delta|$$ is equal to ___________.</p> | [] | null | 32 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsjwcdv7/6c17fcea-c801-4445-8b27-227c5944b078/0d75d830-ca2d-11ee-8854-3b5a6c9e9092/file-6y3zli1lsjwcdv8.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lsjwcdv7/6c17fcea-c801-4445-8b27-227c5944b078/0d75d830-ca2d-11ee... | integer | jee-main-2024-online-31st-january-evening-shift | 8,529 |
jaoe38c1lse4xwyu | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | <p>Let $$\alpha, \beta, \gamma, \delta \in \mathbb{Z}$$ and let $$A(\alpha, \beta), B(1,0), C(\gamma, \delta)$$ and $$D(1,2)$$ be the vertices of a parallelogram $$\mathrm{ABCD}$$. If $$A B=\sqrt{10}$$ and the points $$\mathrm{A}$$ and $$\mathrm{C}$$ lie on the line $$3 y=2 x+1$$, then $$2(\alpha+\beta+\gamma+\delta)$$... | [{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "5"}, {"identifier": "C", "content": "12"}, {"identifier": "D", "content": "10"}] | ["A"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsnjikpx/f32556f0-5b8e-46de-b6e9-9f078c18e4de/05872950-cc2e-11ee-b20d-39b621d226e3/file-6y3zli1lsnjikpy.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lsnjikpx/f32556f0-5b8e-46de-b6e9-9f078c18e4de/05872950-cc2e-11ee... | mcq | jee-main-2024-online-31st-january-morning-shift | 8,530 |
1lsg96zxo | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | <p>A line passing through the point $$\mathrm{A}(9,0)$$ makes an angle of $$30^{\circ}$$ with the positive direction of $$x$$-axis. If this line is rotated about A through an angle of $$15^{\circ}$$ in the clockwise direction, then its equation in the new position is :</p> | [{"identifier": "A", "content": "$$\\frac{y}{\\sqrt{3}+2}+x=9$$\n"}, {"identifier": "B", "content": "$$\\frac{x}{\\sqrt{3}+2}+y=9$$\n"}, {"identifier": "C", "content": "$$\\frac{x}{\\sqrt{3}-2}+y=9$$\n"}, {"identifier": "D", "content": "$$\\frac{y}{\\sqrt{3}-2}+x=9$$"}] | ["D"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsqnkh52/7d853a24-3c49-423c-95b3-38a63c841203/38d89660-cde4-11ee-a0d3-7b75c4537559/file-6y3zli1lsqnkh53.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lsqnkh52/7d853a24-3c49-423c-95b3-38a63c841203/38d89660-cde4-11ee... | mcq | jee-main-2024-online-30th-january-morning-shift | 8,531 |
luy6z55g | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | <p>A ray of light coming from the point $$\mathrm{P}(1,2)$$ gets reflected from the point $$\mathrm{Q}$$ on the $$x$$-axis and then passes through the point $$R(4,3)$$. If the point $$S(h, k)$$ is such that $$P Q R S$$ is a parallelogram, then $$hk^2$$ is equal to:</p> | [{"identifier": "A", "content": "60"}, {"identifier": "B", "content": "70"}, {"identifier": "C", "content": "80"}, {"identifier": "D", "content": "90"}] | ["B"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lw37w9ht/b3e9aa4d-c4a4-4387-bd7f-990e7790d425/c5298810-1031-11ef-b980-477f779c8c59/file-1lw37w9hu.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lw37w9ht/b3e9aa4d-c4a4-4387-bd7f-990e7790d425/c5298810-1031-11ef-b980-477f779c8c59... | mcq | jee-main-2024-online-9th-april-morning-shift | 8,532 |
lvc57b6s | maths | straight-lines-and-pair-of-straight-lines | various-forms-of-straight-line | <p>Let a variable line of slope $$m>0$$ passing through the point $$(4,-9)$$ intersect the coordinate axes at the points $$A$$ and $$B$$. The minimum value of the sum of the distances of $$A$$ and $$B$$ from the origin is</p> | [{"identifier": "A", "content": "30"}, {"identifier": "B", "content": "15"}, {"identifier": "C", "content": "10"}, {"identifier": "D", "content": "25"}] | ["D"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwd2p5t3/fdf80bf0-c51d-4c22-865c-bd4b3e3d59be/487f8460-159d-11ef-aabb-5de744d2ad81/file-1lwd2p5t4.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lwd2p5t3/fdf80bf0-c51d-4c22-865c-bd4b3e3d59be/487f8460-159d-11ef-aabb-5de744d2ad81... | mcq | jee-main-2024-online-6th-april-morning-shift | 8,533 |
IG2r7SisPd2bkNPztoE7U | maths | trigonometric-functions-and-equations | general-solution-and-principal-solution-of-the-equation | If 0 $$ \le $$ x < $${\pi \over 2}$$, then the number of values of x for which sin x $$-$$ sin 2x + sin 3x = 0, is : | [{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "4"}, {"identifier": "D", "content": "2"}] | ["D"] | null | sin x $$-$$ sin 2x + sin 3x = 0 $$x \in \left[ {0,{\pi \over 2}} \right)$$
<br><br>$$ \Rightarrow $$ (sin3x + sinx) $$-$$ sin2x = 0
<br><br>$$ \Rightarrow $$ 2sin2x.cos2x $$-$$ sin2x = 0
<br><br>$$ \Rightarrow $$ sin2x (2cosx $$-$$ 1) = 0
... | mcq | jee-main-2019-online-9th-january-evening-slot | 8,534 |
lsamiw6x | maths | trigonometric-functions-and-equations | solving-trigonometric-equations | The number of solutions of the equation $4 \sin ^2 x-4 \cos ^3 x+9-4 \cos x=0 ; x \in[-2 \pi, 2 \pi]$ is : | [{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "2"}] | ["A"] | null | <p>We start by recognizing that $
\sin^2 x +
\cos^2 x = 1$. Substituting $
\sin^2 x = 1 -
\cos^2 x$ into the original equation gives:</p>
<p>$$
4(1 -
\cos^2 x) - 4
\cos^3 x + 9 - 4
\cos x = 0
$$</p>
<p>Rearranging and simplifying this equation, we have:</p>
<p>$$
4 - 4
\cos^2 x - 4
\cos^3 x + 9 - 4
\cos x = 0$$</p>
... | mcq | jee-main-2024-online-1st-february-evening-shift | 8,536 |
jaoe38c1lscnfni9 | maths | trigonometric-functions-and-equations | solving-trigonometric-equations | <p>If $$2 \tan ^2 \theta-5 \sec \theta=1$$ has exactly 7 solutions in the interval $$\left[0, \frac{n \pi}{2}\right]$$, for the least value of $$n \in \mathbf{N}$$, then $$\sum_\limits{k=1}^n \frac{k}{2^k}$$ is equal to:</p> | [{"identifier": "A", "content": "$$\\frac{1}{2^{14}}\\left(2^{15}-15\\right)$$\n"}, {"identifier": "B", "content": "$$1-\\frac{15}{2^{13}}$$\n"}, {"identifier": "C", "content": "$$\\frac{1}{2^{15}}\\left(2^{14}-14\\right)$$\n"}, {"identifier": "D", "content": "$$\\frac{1}{2^{13}}\\left(2^{14}-15\\right)$$"}] | ["D"] | null | <p>$$\begin{aligned}
& 2 \tan ^2 \theta-5 \sec \theta-1=0 \\
& \Rightarrow 2 \sec ^2 \theta-5 \sec \theta-3=0 \\
& \Rightarrow(2 \sec \theta+1)(\sec \theta-3)=0 \\
& \Rightarrow \sec \theta=-\frac{1}{2}, 3 \\
& \Rightarrow \cos \theta=-2, \frac{1}{3} \\
& \Rightarrow \cos \theta=\frac{1}{3}
\end{aligned}$$</p>
<p>For 7... | mcq | jee-main-2024-online-27th-january-evening-shift | 8,537 |
jaoe38c1lseydb0d | maths | trigonometric-functions-and-equations | solving-trigonometric-equations | <p>If $$\alpha,-\frac{\pi}{2}<\alpha<\frac{\pi}{2}$$ is the solution of $$4 \cos \theta+5 \sin \theta=1$$, then the value of $$\tan \alpha$$ is</p> | [{"identifier": "A", "content": "$$\\frac{10-\\sqrt{10}}{12}$$\n"}, {"identifier": "B", "content": "$$\\frac{\\sqrt{10}-10}{6}$$\n"}, {"identifier": "C", "content": "$$\\frac{\\sqrt{10}-10}{12}$$\n"}, {"identifier": "D", "content": "$$\\frac{10-\\sqrt{10}}{6}$$"}] | ["C"] | null | <p>$$4+5 \tan \theta=\sec \theta$$</p>
<p>Squaring : $$24 \tan ^2 \theta+40 \tan \theta+15=0$$</p>
<p>$$\tan \theta=\frac{-10 \pm \sqrt{10}}{12}$$</p>
<p>and $$\tan \theta=-\left(\frac{10+\sqrt{10}}{12}\right)$$ is Rejected.</p>
<p>(3) is correct.</p> | mcq | jee-main-2024-online-29th-january-morning-shift | 8,538 |
jaoe38c1lsfkwdj6 | maths | trigonometric-functions-and-equations | solving-trigonometric-equations | <p>The sum of the solutions $$x \in \mathbb{R}$$ of the equation $$\frac{3 \cos 2 x+\cos ^3 2 x}{\cos ^6 x-\sin ^6 x}=x^3-x^2+6$$ is</p> | [{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "0"}, {"identifier": "D", "content": "$$-$$1"}] | ["D"] | null | <p>$$\begin{aligned}
& \frac{3 \cos 2 x+\cos ^3 2 x}{\cos ^6 x-\sin ^6 x}=x^3-x^2+6 \\
& \Rightarrow \frac{\cos 2 x\left(3+\cos ^2 2 x\right)}{\cos 2 x\left(1-\sin ^2 x \cos ^2 x\right)}=x^3-x^2+6 \\
& \Rightarrow \frac{4\left(3+\cos ^2 2 x\right)}{\left(4-\sin ^2 2 x\right)}=x^3-x^2+6 \\
& \Rightarrow \frac{4\left(3+\... | mcq | jee-main-2024-online-29th-january-evening-shift | 8,539 |
lv2erh14 | maths | trigonometric-functions-and-equations | solving-trigonometric-equations | <p>Let $$S=\left\{\sin ^2 2 \theta:\left(\sin ^4 \theta+\cos ^4 \theta\right) x^2+(\sin 2 \theta) x+\left(\sin ^6 \theta+\cos ^6 \theta\right)=0\right.$$ has real roots $$\}$$. If $$\alpha$$ and $$\beta$$ be the smallest and largest elements of the set $$S$$, respectively, then $$3\left((\alpha-2)^2+(\beta-1)^2\right)$... | [] | null | 4 | <p>For real roots</p>
<p>$$\begin{aligned}
& D \geq 0 \\
& \sin ^2 2 \theta \geq 4\left(\sin ^4 \theta+\cos ^4 \theta\right)\left(\sin ^6 \theta+\cos ^6 \theta\right)
\end{aligned}$$</p>
<p>Put $$\sin ^2 2 \theta=t$$</p>
<p>$$\begin{aligned}
& \Rightarrow t \geq 4\left(1-\frac{t}{2}\right)\left(1-\frac{3 t}{4}\right) \... | integer | jee-main-2024-online-4th-april-evening-shift | 8,542 |
3BKJ6l1wQpkLN7tB | maths | trigonometric-ratio-and-identites | addition-and-subtraction-formula | Let $$\alpha ,\,\beta $$ be such that $$\pi < \alpha - \beta < 3\pi $$.
<br/>If $$sin{\mkern 1mu} \alpha + \sin \beta = - {{21} \over {65}}$$ and $$\cos \alpha + \cos \beta = - {{27} \over {65}}$$ then the value of $$\cos {{\alpha - \beta } \over 2}$$ : | [{"identifier": "A", "content": "$${{ - 6} \\over {65}}\\,\\,$$ "}, {"identifier": "B", "content": "$${3 \\over {\\sqrt {130} }}$$ "}, {"identifier": "C", "content": "$${6 \\over {65}}$$ "}, {"identifier": "D", "content": "$$ - {3 \\over {\\sqrt {130} }}$$"}] | ["D"] | null | Given $$sin{\mkern 1mu} \alpha + \sin \beta = - {{21} \over {65}}$$ .........(1) <br><br>and $$\cos \alpha + \cos \beta = - {{27} \over {65}}$$ ........(2)
<br><br>Square and add (1) and (2) you will get
<br><br>$$2\left( {1 + \cos \alpha \cos \beta + \sin \alpha \sin \beta } \right)$$$$ = {{{{\left( {21} \... | mcq | aieee-2004 | 8,544 |
hd48oCskZmQcbGFf | maths | trigonometric-ratio-and-identites | addition-and-subtraction-formula | Let <b>A</b> and <b>B</b> denote the statements
<p><b>A</b>: $$\cos \alpha + \cos \beta + \cos \gamma = 0$$</p>
<p><b>B</b>: $$\sin \alpha + \sin \beta + \sin \gamma = 0$$</p>
<p>If $$\cos \left( {\beta - \gamma } \right) + \cos \left( {\gamma - \alpha } \right) + \cos \left( {\alpha - \beta } \right) = - {3... | [{"identifier": "A", "content": "<b>A</b> is false and <b>B</b> is true "}, {"identifier": "B", "content": "both <b>A</b> and <b>B</b> are true "}, {"identifier": "C", "content": "both <b>A</b> and <b>B</b> are false "}, {"identifier": "D", "content": "<b>A</b> is true and <b>B</b> is false"}] | ["B"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265710/exam_images/vm4amwtny6bublz2trzy.webp" loading="lazy" alt="AIEEE 2009 Mathematics - Trigonometric Ratio and Identites Question 48 English Explanation"> | mcq | aieee-2009 | 8,545 |
qWyukXL9UGk1iVvA | maths | trigonometric-ratio-and-identites | addition-and-subtraction-formula | Let $$\cos \left( {\alpha + \beta } \right) = {4 \over 5}$$ and $$\sin \,\,\,\left( {\alpha - \beta } \right) = {5 \over {13}},$$ where $$0 \le \alpha ,\,\beta \le {\pi \over 4}.$$
<br/>Then $$tan\,2\alpha $$ =<br/> | [{"identifier": "A", "content": "$${56 \\over 33}$$"}, {"identifier": "B", "content": "$${19 \\over 12}$$ "}, {"identifier": "C", "content": "$${20 \\over 7}$$"}, {"identifier": "D", "content": "$${25 \\over 16}$$ "}] | ["A"] | null | $$\cos \left( {\alpha + \beta } \right) = {4 \over 5} \Rightarrow \tan \left( {\alpha + \beta } \right) = {3 \over 4}$$
<br><br>$$\sin \left( {\alpha - \beta } \right) = {5 \over {13}} \Rightarrow \tan \left( {\alpha - \beta } \right) = {5 \over {12}}$$
<br><br>$$\tan 2\alpha = \tan \left[ {\left( {\alpha + \beta... | mcq | aieee-2010 | 8,546 |
bphMWKhkG9CXaybI3b30Z | maths | trigonometric-ratio-and-identites | addition-and-subtraction-formula | If cos($$\alpha $$ + $$\beta $$) = 3/5 ,sin ( $$\alpha $$ - $$\beta $$) = 5/13 and
0 < $$\alpha , \beta$$ < $$\pi \over 4$$, then tan(2$$\alpha $$) is equal to : | [{"identifier": "A", "content": "21/16"}, {"identifier": "B", "content": "63/52"}, {"identifier": "C", "content": "33/52"}, {"identifier": "D", "content": "63/16"}] | ["D"] | null | Given $$0 < \alpha < {\pi \over 4}$$
<br><br>and $$0 < \beta < {\pi \over 4}$$
<br><br>$$ \therefore $$ $$0 > - \beta > - {\pi \over 4}$$
<br><br>$$ \therefore $$ $$0 < \alpha + \beta < {\pi \over 2}$$
<br><br>and $$ - {\pi \over 4} < \alpha - \beta < {\pi \over 4}$$
<br><br... | mcq | jee-main-2019-online-8th-april-morning-slot | 8,547 |
BsuLEX4QMoaT0JZXqP1klt7sbsa | maths | trigonometric-ratio-and-identites | addition-and-subtraction-formula | If 0 < x, y < $$\pi$$ and cosx + cosy $$-$$ cos(x + y) = $${3 \over 2}$$, then sinx + cosy is equal to : | [{"identifier": "A", "content": "$${{1 + \\sqrt 3 } \\over 2}$$"}, {"identifier": "B", "content": "$${{1 \\over 2}}$$"}, {"identifier": "C", "content": "$${{\\sqrt 3 } \\over 2}$$"}, {"identifier": "D", "content": "$${{1 - \\sqrt 3 } \\over 2}$$"}] | ["A"] | null | $$2\cos \left( {{{x + y} \over 2}} \right)\cos \left( {{{x - y} \over 2}} \right) - \left[ {2{{\cos }^2}\left( {{{x + y} \over 2}} \right) - 1} \right] = {3 \over 2}$$<br><br>$$2\cos \left( {{{x + y} \over 2}} \right)\left[ {\cos \left( {{{x - y} \over 2}} \right) - \cos \left( {{{x + y} \over 2}} \right)} \right] = {1... | mcq | jee-main-2021-online-25th-february-evening-slot | 8,548 |
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