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1ktbiovly | maths | circle | basic-theorems-of-a-circle | The locus of a point, which moves such that the sum of squares of its distances from the points (0, 0), (1, 0), (0, 1), (1, 1) is 18 units, is a circle of diameter d. Then d<sup>2</sup> is equal to _____________. | [] | null | 16 | Let point P(x, y)
<br><br>A(0, 0), B(1, 0), C(0, 1), D(1, 1)
<br><br>(PA)<sup>2</sup> + (PB)<sup>2</sup> + (PC)<sup>2</sup> + (PD)<sup>2</sup> = 18
<br><br>$${x^2} + {y^2} + {x^2} + {(y - 1)^2} + {(x - 1)^2} + {y^2} + {(x - 1)^2} + {(y - 1)^2}$$ = 18<br><br>$$ \Rightarrow 4({x^2} + {y^2}) - 4y - 4x = 14$$<br><br>$$ \Ri... | integer | jee-main-2021-online-26th-august-morning-shift | 5,202 |
1ktg2remf | maths | circle | basic-theorems-of-a-circle | Let Z be the set of all integers,<br/><br/>$$A = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {y^2} \le 4\} $$<br/><br/>$$B = \{ (x,y) \in Z \times Z:{x^2} + {y^2} \le 4\} $$<br/><br/>$$C = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {(y - 2)^2} \le 4\} $$<br/><br/>If the total number of relation from A $$\cap$$ B to A $$\cap$$ C i... | [{"identifier": "A", "content": "16"}, {"identifier": "B", "content": "25"}, {"identifier": "C", "content": "49"}, {"identifier": "D", "content": "9"}] | ["B"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265832/exam_images/dtgcpzs2ufin73ryqlsn.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264735/exam_images/vsklxgqplszmejwhrtui.webp"><img src="https://res.c... | mcq | jee-main-2021-online-27th-august-evening-shift | 5,203 |
1l54bapov | maths | circle | basic-theorems-of-a-circle | <p>Let a triangle ABC be inscribed in the circle $${x^2} - \sqrt 2 (x + y) + {y^2} = 0$$ such that $$\angle BAC = {\pi \over 2}$$. If the length of side AB is $$\sqrt 2 $$, then the area of the $$\Delta$$ABC is equal to :</p> | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "$$\\left( {\\sqrt 6 + \\sqrt 3 } \\right)/2$$"}, {"identifier": "C", "content": "$$\\left( {3 + \\sqrt 3 } \\right)/4$$"}, {"identifier": "D", "content": "$$\\left( {\\sqrt 6 + 2\\sqrt 3 } \\right)/4$$"}] | ["A"] | null | <p>Note:</p>
<p>For equation of circle $${x^2} + {y^2} + 2gx + 2fy + c = 0$$, center is $$( - g,\, - f)$$ and radius $$r = \sqrt {{g^2} + {f^2} - c} $$</p>
<p>Given,</p>
<p>equation of circle is</p>
<p>$${x^2} - \sqrt 2 (x + y) + {y^2} = 0$$</p>
<p>$$ \Rightarrow {x^2} + {y^2} - \sqrt 2 x - \sqrt 2 y = 0$$</p>
<p>$$ \R... | mcq | jee-main-2022-online-29th-june-evening-shift | 5,204 |
1l57p6r2o | maths | circle | basic-theorems-of-a-circle | <p>A rectangle R with end points of one of its sides as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x $$-$$ y + 4 = 0, then the area of R is ____________.</p> | [] | null | 16 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5qa5m7e/7e601531-e009-4eeb-99e6-ea8138a03412/cb3a40a0-0656-11ed-903e-c9687588b3f3/file-1l5qa5m7f.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5qa5m7e/7e601531-e009-4eeb-99e6-ea8138a03412/cb3a40a0-0656-11ed-903e-c9687588b3f3... | integer | jee-main-2022-online-27th-june-morning-shift | 5,205 |
1l6rg5oz4 | maths | circle | basic-theorems-of-a-circle | <p>$$\text { Let } S=\left\{(x, y) \in \mathbb{N} \times \mathbb{N}: 9(x-3)^{2}+16(y-4)^{2} \leq 144\right\}$$ and
$$T=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}:(x-7)^{2}+(y-4)^{2} \leq 36\right\}$$. Then $$n(S \cap T)$$ is equal to __________.</p> | [] | null | 27 | $S=\left\{(x, y) \in \mathbb{N} \times \mathbb{N}: \frac{(x-3)^{2}}{16}+\frac{(y-4)^{2}}{9} \leq 1\right\}$ <br><br>represents all the integral points inside <br><br>and on the ellipse $\frac{(x-3)^{2}}{16}+\frac{(y-4)^{2}}{9}=1$, in first quadrant.
<br><br>
and $T=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}:(x-7)^{... | integer | jee-main-2022-online-29th-july-evening-shift | 5,206 |
1lh00nw5i | maths | circle | basic-theorems-of-a-circle | <p>Consider a circle $$C_{1}: x^{2}+y^{2}-4 x-2 y=\alpha-5$$. Let its mirror image in the line $$y=2 x+1$$ be another circle $$C_{2}: 5 x^{2}+5 y^{2}-10 f x-10 g y+36=0$$. Let $$r$$ be the radius of $$C_{2}$$. Then $$\alpha+r$$ is equal to _________.</p> | [] | null | 2 | We have,
<br/><br/>$$
\begin{aligned}
& C_1: x^2+y^2-4 x-2 y=\alpha-5 \\\\
& C_1:(x-2)^2+(y-1)^5-5=\alpha-5 \\\\
& C_1:(x-2)^2+(y-1)^2=(\sqrt{\alpha})^2
\end{aligned}
$$
<br/><br/>So, centre and radius of $C_1$ are $(2,1)$ and $\sqrt{\alpha}$ respectively
<br/><br/>Now, image of $(2,1)$ along the line $y=2 x+1$ is,
<br... | integer | jee-main-2023-online-8th-april-morning-shift | 5,209 |
1lh23ruwz | maths | circle | basic-theorems-of-a-circle | <p>Let the point $$(p, p+1)$$ lie inside the region $$E=\left\{(x, y): 3-x \leq y \leq \sqrt{9-x^{2}}, 0 \leq x \leq 3\right\}$$. If the set of all values of $$\mathrm{p}$$ is the interval $$(a, b)$$, then $$b^{2}+b-a^{2}$$ is equal to ___________.</p> | [] | null | 3 | Given region,
<br/><br/>$$
E=\left\{(x, y): 3-x \leq y \leq \sqrt{9-x^2}, 0 \leq x \leq 3\right\}
$$
<br/><br/>Since, point $(p, p+1)$ lie on line $y=x+1$
<br/><br/>$\therefore$ Point of intersection of $y=x+1$ and $y=3-x$
<br/><br/>i.e.,
$x+1=3-x$
<br/><br/>$\Rightarrow$ $2 x=2 \Rightarrow x=1$
<br/><br/>and $y=2$
<b... | integer | jee-main-2023-online-6th-april-morning-shift | 5,210 |
luy6z5cm | maths | circle | basic-theorems-of-a-circle | <p>Let a circle passing through $$(2,0)$$ have its centre at the point $$(\mathrm{h}, \mathrm{k})$$. Let $$(x_{\mathrm{c}}, y_{\mathrm{c}})$$ be the point of intersection of the lines $$3 x+5 y=1$$ and $$(2+\mathrm{c}) x+5 \mathrm{c}^2 y=1$$. If $$\mathrm{h}=\lim _\limits{\mathrm{c} \rightarrow 1} x_{\mathrm{c}}$$ and ... | [{"identifier": "A", "content": "$$5 x^2+5 y^2-4 x-2 y-12=0$$\n"}, {"identifier": "B", "content": "$$25 x^2+25 y^2-20 x+2 y-60=0$$\n"}, {"identifier": "C", "content": "$$25 x^2+25 y^2-2 x+2 y-60=0$$\n"}, {"identifier": "D", "content": "$$5 x^2+5 y^2-4 x+2 y-12=0$$"}] | ["B"] | null | <p>$$\begin{aligned}
& 3 x+5 y=1 \\
& (2+c) x+5 c^2 y=1 \\
& 3 c^2 x+5 c^2 y=c^2
\end{aligned}$$</p>
<p>Subtracting</p>
<p>$$\begin{aligned}
& \left(2+c-3 c^2\right) x=1-c^2 \\
& x_c=\frac{1-c^2}{2+c-3 c^2}=\frac{(1-c)(1+c)}{(1-c)(3 c+2)}=\frac{c+1}{3 c+2} \\
& y=\frac{1-3 x}{5}=\frac{1-3\left(\frac{c+1}{3 c+2}\right)}... | mcq | jee-main-2024-online-9th-april-morning-shift | 5,211 |
lv0vxc6v | maths | circle | basic-theorems-of-a-circle | <p>A square is inscribed in the circle $$x^2+y^2-10 x-6 y+30=0$$. One side of this square is parallel to $$y=x+3$$. If $$\left(x_i, y_i\right)$$ are the vertices of the square, then $$\Sigma\left(x_i^2+y_i^2\right)$$ is equal to:</p> | [{"identifier": "A", "content": "152"}, {"identifier": "B", "content": "148"}, {"identifier": "C", "content": "156"}, {"identifier": "D", "content": "160"}] | ["A"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwk74s6l/678de8fc-2a12-44e3-8b6b-dd8440b79fd7/1bfb85d0-1988-11ef-a7bd-376696e028ce/file-1lwk74s6m.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lwk74s6l/678de8fc-2a12-44e3-8b6b-dd8440b79fd7/1bfb85d0-1988-11ef-a7bd-376696e028ce... | mcq | jee-main-2024-online-4th-april-morning-shift | 5,212 |
lv3ve68b | maths | circle | basic-theorems-of-a-circle | <p>If the image of the point $$(-4,5)$$ in the line $$x+2 y=2$$ lies on the circle $$(x+4)^2+(y-3)^2=r^2$$, then $$r$$ is equal to:</p> | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "4"}, {"identifier": "D", "content": "1"}] | ["A"] | null | <p>$$\begin{aligned}
& \frac{x+4}{1}=\frac{y-5}{2}=\frac{-2(4)}{5} \\
& \Rightarrow \quad x=-4-\frac{8}{5}=-\frac{28}{5}, y=5-\frac{16}{5}=\frac{9}{5} \\
& \therefore \quad \text { Image is }\left(\frac{-28}{5}, \frac{9}{5}\right)
\end{aligned}$$</p>
<p>Image lies on circle $$(x+4)^2+(y-3)^2=r^2$$</p>
<p>$$\begin{align... | mcq | jee-main-2024-online-8th-april-evening-shift | 5,213 |
lv9s1zub | maths | circle | basic-theorems-of-a-circle | <p>Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies :</p> | [{"identifier": "A", "content": "$$\\mathrm{r}=1$$\n"}, {"identifier": "B", "content": "$$2 \\mathrm{r}^2-4 \\mathrm{r}+1=0$$\n"}, {"identifier": "C", "content": "$$2 \\mathrm{r}^2-8 \\mathrm{r}+7=0$$\n"}, {"identifier": "D", "content": "$$\\mathrm{r}^2-8 \\mathrm{r}+8=0$$"}] | ["D"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lweo69as/998331bc-73f9-46f0-9260-389c06031bce/0c501540-167e-11ef-9070-f523f4c6bd4b/file-1lweo69at.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lweo69as/998331bc-73f9-46f0-9260-389c06031bce/0c501540-167e-11ef-9070-f523f4c6bd4b... | mcq | jee-main-2024-online-5th-april-evening-shift | 5,214 |
lvc57bdz | maths | circle | basic-theorems-of-a-circle | <p>A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are $$m$$ and $$n$$, respectively, then $$m+n^2$$ is equal to</p> | [{"identifier": "A", "content": "408"}, {"identifier": "B", "content": "414"}, {"identifier": "C", "content": "312"}, {"identifier": "D", "content": "396"}] | ["A"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwd0dvt3/ce397cb2-87aa-4135-a838-45d211acf529/3c9d7570-1594-11ef-88c8-4b364e13ab15/file-1lwd0dvt4.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lwd0dvt3/ce397cb2-87aa-4135-a838-45d211acf529/3c9d7570-1594-11ef-88c8-4b364e13ab15... | mcq | jee-main-2024-online-6th-april-morning-shift | 5,215 |
1l5bb5tdb | maths | circle | chord-of-contact | <p>Let a circle C : (x $$-$$ h)<sup>2</sup> + (y $$-$$ k)<sup>2</sup> = r<sup>2</sup>, k > 0, touch the x-axis at (1, 0). If the line x + y = 0 intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of h + k + r is equal to ___________.</p> | [] | null | 7 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5v64hz0/fe62d372-86ba-43d3-b376-3bcc7cdf6f30/f2e103c0-0906-11ed-a790-b11fa70c8a36/file-1l5v64hz1.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5v64hz0/fe62d372-86ba-43d3-b376-3bcc7cdf6f30/f2e103c0-0906-11ed-a790-b11fa70c8a36... | integer | jee-main-2022-online-24th-june-evening-shift | 5,216 |
1l6rfz5ti | maths | circle | chord-of-contact | <p>Let $$A B$$ be a chord of length 12 of the circle $$(x-2)^{2}+(y+1)^{2}=\frac{169}{4}$$. If tangents drawn to the circle at points $$A$$ and $$B$$ intersect at the point $$P$$, then five times the distance of point $$P$$ from chord $$A B$$ is equal to __________.</p> | [] | null | 72 | Here $A M=B M=6$
<br><br>$$
O M=\sqrt{\left(\frac{13}{2}\right)^{2}-6^{2}}=\frac{5}{2}
$$
<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7z3hbu0/c0ab874c-f40a-4393-abd9-7c5a221a294e/2d9f8370-32c8-11ed-a433-8d18e842a53d/file-1l7z3hbu1.png?format=png" data-orsrc="https://app-content.cdn.examgo... | integer | jee-main-2022-online-29th-july-evening-shift | 5,217 |
1ldsuhn1h | maths | circle | chord-of-contact | <p>Let the tangents at the points $$A(4,-11)$$ and $$B(8,-5)$$ on the circle $$x^{2}+y^{2}-3 x+10 y-15=0$$, intersect at the point $$C$$. Then the radius of the circle, whose centre is $$C$$ and the line joining $$A$$ and $$B$$ is its tangent, is equal to :</p> | [{"identifier": "A", "content": "$$\\frac{2\\sqrt{13}}{3}$$"}, {"identifier": "B", "content": "$$\\frac{3\\sqrt{3}}{4}$$"}, {"identifier": "C", "content": "$$\\sqrt{13}$$"}, {"identifier": "D", "content": "$$2\\sqrt{13}$$"}] | ["A"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1ldt0172l/c56cbb61-8f00-4972-aa26-7f1c65ed7300/4a06dfd0-a637-11ed-8501-8d588d737388/file-1ldt0174e.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1ldt0172l/c56cbb61-8f00-4972-aa26-7f1c65ed7300/4a06dfd0-a637-11ed-8501-8d588d737388... | mcq | jee-main-2023-online-29th-january-morning-shift | 5,218 |
lsapr1j6 | maths | circle | chord-of-contact | Let $C: x^2+y^2=4$ and $C^{\prime}: x^2+y^2-4 \lambda x+9=0$ be two circles. If the set of all values of $\lambda$ so that the circles $\mathrm{C}$ and $\mathrm{C}$ intersect at two distinct points, is $\mathrm{R}-[\mathrm{a}, \mathrm{b}]$, then the point $(8 \mathrm{a}+12,16 \mathrm{~b}-20)$ lies on the curve : | [{"identifier": "A", "content": "$x^2+2 y^2-5 x+6 y=3$"}, {"identifier": "B", "content": "$5 x^2-y=-11$"}, {"identifier": "C", "content": "$x^2-4 y^2=7$"}, {"identifier": "D", "content": "$6 x^2+y^2=42$"}] | ["D"] | null | $\begin{aligned} & C: x^2+y^2=4 \Rightarrow C(0,0), r_1=2 \\\\ & C^{\prime}: x^2+y^2-4 \lambda x+9=0 \Rightarrow C^{\prime}(2 \lambda, 0), r_2=\sqrt{4 \lambda^2-9} \\\\ & \left|r_1-r_2\right| < C C^{\prime} < \left|r_1+r_2\right| \\\\ & \left|2-\sqrt{4 \lambda^2-9}\right|<|2 \lambda|<2+\sqrt{4 \lambda^2-9} \\\\ & |2 \l... | mcq | jee-main-2024-online-1st-february-morning-shift | 5,219 |
jaoe38c1lse5ofy4 | maths | circle | chord-of-contact | <p>If one of the diameters of the circle $$x^2+y^2-10 x+4 y+13=0$$ is a chord of another circle $$\mathrm{C}$$, whose center is the point of intersection of the lines $$2 x+3 y=12$$ and $$3 x-2 y=5$$, then the radius of the circle $$\mathrm{C}$$ is :</p> | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "3$$\\sqrt2$$"}, {"identifier": "C", "content": "6"}, {"identifier": "D", "content": "$$\\sqrt{20}$$"}] | ["C"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsnjrugw/97f6ed68-0662-482a-8e94-7842336e50a8/07576000-cc2f-11ee-b20d-39b621d226e3/file-6y3zli1lsnjrugx.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lsnjrugw/97f6ed68-0662-482a-8e94-7842336e50a8/07576000-cc2f-11ee... | mcq | jee-main-2024-online-31st-january-morning-shift | 5,220 |
XkwXCuYHq4WaJyZN | maths | circle | chord-with-a-given-middle-point | The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle having area as 154 sq. units. Then the equation of the circle is : | [{"identifier": "A", "content": "$${x^2}\\, + \\,{y^2} - \\,2x\\, + \\,2y\\,\\, = \\,62$$ "}, {"identifier": "B", "content": "$${x^2}\\, + \\,{y^2} + \\,2x\\, - \\,2y\\,\\, = \\,62$$ "}, {"identifier": "C", "content": "$${x^2}\\, + \\,{y^2} + \\,2x\\, - \\,2y\\,\\, = \\,47$$"}, {"identifier": "D", "content": "$${x^2}\\... | ["D"] | null | $$\pi {r^2} = 154 \Rightarrow r = 7$$
<br><br>For center on solving equation
<br><br>$$2x - 3y = 5\& 3x - 4y = 7$$
<br><br>we get $$x = 1,\,y = - 1$$
<br><br>$$\therefore$$ center $$=(1,-1)$$
<br><br>Equation of circle,
<br><br>$${\left( {x - 1} \right)^2} + {\left( {y + 1} \right)^2} = {7^2}$$
<br><br>$${x^2... | mcq | aieee-2003 | 5,222 |
gy3TyrWxXDlyzSzo | maths | circle | chord-with-a-given-middle-point | If the lines 2x + 3y + 1 + 0 and 3x - y - 4 = 0 lie along diameter of a circle of circumference $$10\,\pi $$, then the equation of the circle is : | [{"identifier": "A", "content": "$${x^2}\\, + \\,{y^2} + \\,2x\\, - \\,2y - \\,23\\,\\, = 0$$ "}, {"identifier": "B", "content": "$${x^2}\\, + \\,{y^2} - \\,2x\\, - \\,2y - \\,23\\,\\, = 0$$ "}, {"identifier": "C", "content": "$${x^2}\\, + \\,{y^2} + \\,2x\\, + \\,2y - \\,23\\,\\, = 0$$ "}, {"identifier": "D", "content... | ["D"] | null | Two diameters are along
<br><br>$$2x+3y+1=0$$ and $$3x-y-4=0$$
<br><br>solving we get center $$(1,-1)$$
<br><br>circumference $$ = 2\pi r = 10\pi $$
<br><br>$$\therefore$$ $$r=5$$.
<br><br>Required circle is, $${\left( {x - 1} \right)^2} + {\left( {y + 1} \right)^2} = {5^2}$$
<br><br>$$ \Rightarrow {x^2} + {y^2} - 2x... | mcq | aieee-2004 | 5,223 |
wDezU4MM76FhKiUT | maths | circle | chord-with-a-given-middle-point | If the pair of lines $$a{x^2} + 2\left( {a + b} \right)xy + b{y^2} = 0$$ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then : | [{"identifier": "A", "content": "$$3{a^2} - 10ab + 3{b^2} = 0$$ "}, {"identifier": "B", "content": "$$3{a^2} - 2ab + 3{b^2} = 0$$ "}, {"identifier": "C", "content": "$$3{a^2} + 10ab + 3{b^2} = 0$$ "}, {"identifier": "D", "content": "$$3{a^2} + 2ab + 3{b^2} = 0$$ "}] | ["D"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265897/exam_images/tcowhr59pgnhld1bijkk.webp" loading="lazy" alt="AIEEE 2005 Mathematics - Circle Question 148 English Explanation">
<br><br>As per question area of one sector $$=3$$ area of another sector
<br><br>$$ \Rightarrow $$ ... | mcq | aieee-2005 | 5,225 |
3U5RrWuDdNdu7DGZ | maths | circle | chord-with-a-given-middle-point | If the lines $$3x - 4y - 7 = 0$$ and $$2x - 3y - 5 = 0$$ are two diameters of a circle of area $$49\pi $$ square units, the equation of the circle is : | [{"identifier": "A", "content": "$$\\,{x^2} + {y^2} + 2x\\, - 2y - 47 = 0\\,$$ "}, {"identifier": "B", "content": "$$\\,{x^2} + {y^2} + 2x\\, - 2y - 62 = 0\\,$$"}, {"identifier": "C", "content": "$${x^2} + {y^2} - 2x\\, + 2y - 62 = 0$$ "}, {"identifier": "D", "content": "$${x^2} + {y^2} - 2x\\, + 2y - 47 = 0$$"}] | ["D"] | null | Point of intersection of $$3x - 4y - 7 = 0$$ and
<br><br>$$2x - 3y - 5 = 0$$ is $$\left( {1, - 1} \right)$$ which is the center of the
<br><br>circle and radius $$=7$$
<br><br>$$\therefore$$ Equation is $${\left( {x - 1} \right)^2} + {\left( {y + 1} \right)^2} = 49$$
<br><br>$$ \Rightarrow {x^2} + {y^2} - 2x + 2y - 4... | mcq | aieee-2006 | 5,226 |
YnQUn55VJcVWDi9o | maths | circle | chord-with-a-given-middle-point | Let $$C$$ be the circle with centre $$(0, 0)$$ and radius $$3$$ units. The equation of the locus of the mid points of the chords of the circle $$C$$ that subtend an angle of $${{2\pi } \over 3}$$ at its center is : | [{"identifier": "A", "content": "$${x^2} + {y^2} = {3 \\over 2}$$ "}, {"identifier": "B", "content": "$${x^2} + {y^2} = 1$$ "}, {"identifier": "C", "content": "$${x^2} + {y^2} = {{27} \\over 4}$$ "}, {"identifier": "D", "content": "$${x^2} + {y^2} = {{9} \\over 4}$$"}] | ["D"] | null | Let $$M\left( {h,k} \right)$$ be the mid point of chord $$AB$$ where
<br><br>$$\angle AOB = {{2\pi } \over 3}$$
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263288/exam_images/uamjsjszwgsvyetzibag.webp" loading="lazy" alt="AIEEE 2006 Mathematics - Circle Question 146 ... | mcq | aieee-2006 | 5,227 |
wr3znUdfJFsVkS7T | maths | circle | chord-with-a-given-middle-point | If one of the diameters of the circle, given by the equation, $${x^2} + {y^2} - 4x + 6y - 12 = 0,$$ is a chord of a circle $$S$$, whose centre is at $$(-3, 2)$$, then the radius of $$S$$ is : | [{"identifier": "A", "content": "$$5$$ "}, {"identifier": "B", "content": "$$10$$"}, {"identifier": "C", "content": "$$5\\sqrt 2 $$ "}, {"identifier": "D", "content": "$$5\\sqrt 3 $$"}] | ["D"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266458/exam_images/taibjlarssmymugj1mse.webp" loading="lazy" alt="JEE Main 2016 (Offline) Mathematics - Circle Question 133 English Explanation">
<br><br>Center of $$S$$ : $$O(-3, 2)$$ center of given circle $$A(2, -3)$$
<br><br>$$... | mcq | jee-main-2016-offline | 5,228 |
Bb1rY9owBIJLvGWfcTGpH | maths | circle | chord-with-a-given-middle-point | If two parallel chords of a circle, having diameter 4units, lie on the opposite sides of the center and subtend angles $${\cos ^{ - 1}}\left( {{1 \over 7}} \right)$$ and sec<sup>$$-$$1</sup> (7) at the center respectivey, then the distance between these chords, is : | [{"identifier": "A", "content": "$${4 \\over {\\sqrt 7 }}$$ "}, {"identifier": "B", "content": "$${8 \\over {\\sqrt 7 }}$$ "}, {"identifier": "C", "content": "$${8 \\over 7}$$ "}, {"identifier": "D", "content": "$${16 \\over 7}$$"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266917/exam_images/e9ndaij1z9oqlqstuhea.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2017 (Online) 8th April Morning Slot Mathematics - Circle Question 126 English Explanation">
<br><br>Since cos... | mcq | jee-main-2017-online-8th-april-morning-slot | 5,230 |
NN0JSJAcrtZgFuxsgmSCP | maths | circle | chord-with-a-given-middle-point | A line drawn through the point P(4, 7) cuts the circle x<sup>2</sup> + y<sup>2</sup> = 9 at the points A and B. Then PA⋅PB is equal to : | [{"identifier": "A", "content": "53"}, {"identifier": "B", "content": "56"}, {"identifier": "C", "content": "74"}, {"identifier": "D", "content": "65"}] | ["B"] | null | P(4, 7). Here, x = 4, y = 7
<br><br>$$ \therefore $$ PA $$ \times $$ PB = PT<sup>2</sup>
<br><br>Also; PT = $$\sqrt {{x^2} + {y^2} - {{\left( {x - y} \right)}^2}} $$
<br><br>$$ \Rightarrow $$ PT = $$\sqrt {16 + 49 - 9} $$ = $$\sqrt {56} $$
<br><br>$$ \... | mcq | jee-main-2017-online-9th-april-morning-slot | 5,231 |
R6YFDH6A7gVn9tALWCeel | maths | circle | chord-with-a-given-middle-point | If the area of an equilateral triangle inscribed in the circle x<sup>2</sup> + y<sup>2</sup>
+ 10x + 12y + c = 0 is $$27\sqrt 3 $$ sq units then c is equal to : | [{"identifier": "A", "content": "20"}, {"identifier": "B", "content": "25"}, {"identifier": "C", "content": "$$-$$ 25"}, {"identifier": "D", "content": "13"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266928/exam_images/dalvkc5ol09xl6tzqcfw.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 10th January Evening Slot Mathematics - Circle Question 118 English Explanation">
<br>$$3\left( ... | mcq | jee-main-2019-online-10th-january-evening-slot | 5,232 |
ezwzoSZQrfXnvICEWi3rsa0w2w9jx5zq07t | maths | circle | chord-with-a-given-middle-point | If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90<sup>o</sup>, then
the length (in cm) of their common chord is : | [{"identifier": "A", "content": "$${{13} \\over 5}$$"}, {"identifier": "B", "content": "$${{60} \\over {13}}$$"}, {"identifier": "C", "content": "$${{120} \\over {13}}$$"}, {"identifier": "D", "content": "$${{13} \\over 2}$$"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264886/exam_images/tuxfhzteere5dc5zztu5.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 12th April Morning Slot Mathematics - Circle Question 103 English Explanation">
C<sub>1</sub>C<s... | mcq | jee-main-2019-online-12th-april-morning-slot | 5,233 |
J6DqCFwbzQxYPjKuAF1klrjlmf3 | maths | circle | chord-with-a-given-middle-point | If one of the diameters of the circle x<sup>2</sup> + y<sup>2</sup> - 2x - 6y + 6 = 0 is a chord of another circle 'C',
whose center is at (2, 1), then its radius is ________. | [] | null | 3 | Circle x<sup>2</sup> + y<sup>2</sup> - 2x - 6y + 6 = 0 has centre
O<sub>1</sub>(1, 3) and radius r
= 2.
<br><br>Let centre O<sub>2</sub>
(2, 1) of required circle and its
radius being r.
<br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266179/exam_images/zp0tmjnahzn7ywtwha3e.webp" style="max-width... | integer | jee-main-2021-online-24th-february-morning-slot | 5,235 |
9DizyxVsY5PYMF4DAQ1kluxkl2d | maths | circle | chord-with-a-given-middle-point | If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x<sup>2</sup> + y<sup>2</sup> = 1 is a circle of radius r, then r is equal to : | [{"identifier": "A", "content": "$${1 \\over 4}$$"}, {"identifier": "B", "content": "$${1 \\over 2}$$"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "$${1 \\over 3}$$"}] | ["B"] | null | Let P(h, k) and point on the circle is (cos$$\theta$$, sin$$\theta$$)<br><br>$$ \therefore $$ $${{3 + \cos \theta } \over 2} = h$$ and $${{2 + \sin \theta } \over 2} = k$$<br><br>cos$$\theta$$ = 2h $$-$$ 3 and sin$$\theta$$ = 2h $$-$$ 2<br><br>Squaring and adding we get<br><br>$${(2h - 3)^2} + {(2h - 2)^2} = 1$$<br><br... | mcq | jee-main-2021-online-26th-february-evening-slot | 5,236 |
1ktbehi2f | maths | circle | chord-with-a-given-middle-point | If a line along a chord of the circle 4x<sup>2</sup> + 4y<sup>2</sup> + 120x + 675 = 0, passes through the point ($$-$$30, 0) and is tangent to the parabola y<sup>2</sup> = 30x, then the length of this chord is : | [{"identifier": "A", "content": "5"}, {"identifier": "B", "content": "7"}, {"identifier": "C", "content": "5$${\\sqrt 3 }$$"}, {"identifier": "D", "content": "3$${\\sqrt 5 }$$"}] | ["D"] | null | Equation of tangent to y<sup>2</sup> = 30 x<br><br>y = mx + $${{30} \over {4m}}$$<br><br>Pass thru ($$-$$30, 0) : a = $$-$$30m + $${{30} \over {4m}}$$ $$\Rightarrow$$ m<sup>2</sup> = 1/4<br><br>$$\Rightarrow$$ m = $${1 \over 2}$$ or m = $$-$$$${1 \over 2}$$<br><br>At m = $${1 \over 2}$$ : y = $${x \over 2}$$ + 15 $$\R... | mcq | jee-main-2021-online-26th-august-morning-shift | 5,237 |
1ktd2kuim | maths | circle | chord-with-a-given-middle-point | A circle C touches the line x = 2y at the point (2, 1) and intersects the circle <br/><br/>C<sub>1</sub> : x<sup>2</sup> + y<sup>2</sup> + 2y $$-$$ 5 = 0 at two points P and Q such that PQ is a diameter of C<sub>1</sub>. Then the diameter of C is : | [{"identifier": "A", "content": "$$7\\sqrt 5 $$"}, {"identifier": "B", "content": "15"}, {"identifier": "C", "content": "$$\\sqrt {285} $$"}, {"identifier": "D", "content": "$$4\\sqrt {15} $$"}] | ["A"] | null | (x $$-$$ 2)<sup>2</sup> + (y $$-$$ 1)<sup>2</sup> + $$\lambda$$(x $$-$$ 2y) = 0<br><br>C : x<sup>2</sup> + y<sup>2</sup> + x($$\lambda$$ $$-$$ 4) + y($$-$$2 $$-$$2$$\lambda$$) + 5 = 0<br><br>C<sub>1</sub> : x<sup>2</sup> + y<sup>2</sup> + 2y $$-$$ 5 = 0<br><br>S<sub>1</sub> $$-$$ S<sub>2</sub> = 0 (Equation of PQ)<br><... | mcq | jee-main-2021-online-26th-august-evening-shift | 5,238 |
1l55j0dpj | maths | circle | chord-with-a-given-middle-point | <p>If one of the diameters of the circle $${x^2} + {y^2} - 2\sqrt 2 x - 6\sqrt 2 y + 14 = 0$$ is a chord of the circle $${(x - 2\sqrt 2 )^2} + {(y - 2\sqrt 2 )^2} = {r^2}$$, then the value of r<sup>2</sup> is equal to ____________.</p> | [] | null | 10 | For $x^{2}+y^{2}-2 \sqrt{2} x-6 \sqrt{2} y+14=0$
<br/><br/>
$$
\text { Radius }=\sqrt{(\sqrt{2})^{2}+(3 \sqrt{2})^{2}-14}=\sqrt{6}
$$
<br/><br/>
$\Rightarrow$ Diameter $=2 \sqrt{6}$
<br/><br/>
If this diameter is chord to
<br/><br/>
$$
\begin{aligned}
&(x-2 \sqrt{2})^{2}+(y-2 \sqrt{2})^{2}=r^{2} \text { then } \\\\
&\R... | integer | jee-main-2022-online-28th-june-evening-shift | 5,239 |
1l5ajotbj | maths | circle | chord-with-a-given-middle-point | <p>Let the abscissae of the two points P and Q be the roots of $$2{x^2} - rx + p = 0$$ and the ordinates of P and Q be the roots of $${x^2} - sx - q = 0$$. If the equation of the circle described on PQ as diameter is $$2({x^2} + {y^2}) - 11x - 14y - 22 = 0$$, then $$2r + s - 2q + p$$ is equal to __________.</p> | [] | null | 7 | <p>Let $$P({x_1},{y_1})$$ & $$Q({x_2},{y_2})$$</p>
<p>$$\therefore$$ Roots of $$2{x^2} - rx + p = 0$$ are $${x_1},\,{x_2}$$</p>
<p>and roots of $${x^2} - sx - q = 0$$ are $${y_1},\,{y_2}$$.</p>
<p>$$\therefore$$ Equation of circle $$ \equiv (x - {x_1})(x - {x_2}) + (y - {y_1})(y - {y_2}) = 0$$</p>
<p>$$ \Rightarrow {x^... | integer | jee-main-2022-online-25th-june-morning-shift | 5,240 |
1l6hyv47p | maths | circle | chord-with-a-given-middle-point | <p>Let the abscissae of the two points $$P$$ and $$Q$$ on a circle be the roots of $$x^{2}-4 x-6=0$$ and the ordinates of $$\mathrm{P}$$ and $$\mathrm{Q}$$ be the roots of $$y^{2}+2 y-7=0$$. If $$\mathrm{PQ}$$ is a diameter of the circle $$x^{2}+y^{2}+2 a x+2 b y+c=0$$, then the value of $$(a+b-c)$$ is _____________.</... | [{"identifier": "A", "content": "12"}, {"identifier": "B", "content": "13"}, {"identifier": "C", "content": "14"}, {"identifier": "D", "content": "16"}] | ["A"] | null | <p>Abscissae of PQ are roots of $${x^2} - 4x - 6 = 0$$</p>
<p>Ordinates of PQ are roots of $${y^2} + 2y - 7 = 0$$</p>
<p>and PQ is diameter</p>
<p>$$\Rightarrow$$ Equation of circle is</p>
<p>$${x^2} + {y^2} - 4x + 2y - 13 = 0$$</p>
<p>But, given $${x^2} + {y^2} + 2ax + 2by + c = 0$$</p>
<p>By comparison $$a = - 2,b =... | mcq | jee-main-2022-online-26th-july-evening-shift | 5,241 |
ldoa8b8u | maths | circle | chord-with-a-given-middle-point | The set of all values of $a^{2}$ for which the line $x+y=0$ bisects two distinct chords drawn from a point $\mathrm{P}\left(\frac{1+a}{2}, \frac{1-a}{2}\right)$ on the circle $2 x^{2}+2 y^{2}-(1+a) x-(1-a) y=0$, is equal to : | [{"identifier": "A", "content": "$(0,4]$"}, {"identifier": "B", "content": "$(4, \\infty)$"}, {"identifier": "C", "content": "$(2,12]$"}, {"identifier": "D", "content": "$(8, \\infty)$"}] | ["D"] | null | $x^{2}+y^{2}-\frac{(1+a) x}{2}-\frac{(1-a) y}{2}=0$
<br><br>Centre $\left(\frac{1+\mathrm{a}}{4}, \frac{1-\mathrm{a}}{4}\right) \Rightarrow(\mathrm{h}, \mathrm{k})$
<br><br>$\mathrm{P}\left(\frac{1+\mathrm{a}}{2}, \frac{1-\mathrm{a}}{2}\right) \Rightarrow(2 \mathrm{h}, 2 \mathrm{k})$
<br><br><img src="https://app-co... | mcq | jee-main-2023-online-31st-january-evening-shift | 5,242 |
1ldwwp6i8 | maths | circle | chord-with-a-given-middle-point | <p>The locus of the mid points of the chords of the circle $${C_1}:{(x - 4)^2} + {(y - 5)^2} = 4$$ which subtend an angle $${\theta _i}$$ at the centre of the circle $$C_1$$, is a circle of radius $$r_i$$. If $${\theta _1} = {\pi \over 3},{\theta _3} = {{2\pi } \over 3}$$ and $$r_1^2 = r_2^2 + r_3^2$$, then $${\theta ... | [{"identifier": "A", "content": "$${\\pi \\over 2}$$"}, {"identifier": "B", "content": "$${\\pi \\over 4}$$"}, {"identifier": "C", "content": "$${{3\\pi } \\over 4}$$"}, {"identifier": "D", "content": "$${\\pi \\over 6}$$"}] | ["A"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1le5ga481/2ad2340b-953f-4942-9834-237efb8eb334/3bee0510-ad10-11ed-a86d-8dfe0389db88/file-1le5ga482.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1le5ga481/2ad2340b-953f-4942-9834-237efb8eb334/3bee0510-ad10-11ed-a86d-8dfe0389db88... | mcq | jee-main-2023-online-24th-january-evening-shift | 5,243 |
1lgxvzvzw | maths | circle | chord-with-a-given-middle-point | <p>A line segment AB of length $$\lambda$$ moves such that the points A and B remain on the periphery of a circle of radius $$\lambda$$. Then the locus of the point, that divides the line segment AB in the ratio 2 : 3, is a circle of radius :</p> | [{"identifier": "A", "content": "$${2 \\over 3}\\lambda $$"}, {"identifier": "B", "content": "$${3 \\over 5}\\lambda $$"}, {"identifier": "C", "content": "$${{\\sqrt {19} } \\over 7}\\lambda $$"}, {"identifier": "D", "content": "$${{\\sqrt {19} } \\over 5}\\lambda $$"}] | ["D"] | null | Given, length of $A B=\lambda$
<br><br>So, $A C=\frac{\lambda}{2}$ and $A M=\frac{2 \lambda}{5}$
<br><br>$$
C M=A C-A M=\frac{\lambda}{2}-\frac{2 \lambda}{5}=\frac{\lambda}{10}
$$
<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lncsenht/b19528f0-d9c0-44df-a496-57a56c3fa6bf/168f1e10-6347-1... | mcq | jee-main-2023-online-10th-april-morning-shift | 5,244 |
lsamkf38 | maths | circle | chord-with-a-given-middle-point | Let the locus of the midpoints of the chords of the circle $x^2+(y-1)^2=1$ drawn from the origin intersect the line $x+y=1$ at $\mathrm{P}$ and $\mathrm{Q}$. Then, the length of $\mathrm{PQ}$ is : | [{"identifier": "A", "content": "$\\frac{1}{2}$"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "$\\frac{1}{\\sqrt{2}}$"}, {"identifier": "D", "content": "$\\sqrt{2}$"}] | ["C"] | null | Let mid-point is $(x, y)$
<br/><br/>$$
\begin{aligned}
& x^2+y^2-2 y=0 \\\\
& x x_1+y y_1-\left(y+y_1\right)=x_1^2+y_1^2-2 y_1
\end{aligned}
$$
<br/><br/>It is passing through origin
<br/><br/>$$
\begin{aligned}
& \text { So, } 0+0-\left(0+y_1\right)=x_1^2+y_1^2-2 y_1 \\\\
& \Rightarrow -y_1=x_1^2+y_1^2-2 y_1 \\\\
& \R... | mcq | jee-main-2024-online-1st-february-evening-shift | 5,245 |
lv2eruuk | maths | circle | chord-with-a-given-middle-point | <p>Let $$\mathrm{C}$$ be a circle with radius $$\sqrt{10}$$ units and centre at the origin. Let the line $$x+y=2$$ intersects the circle $$\mathrm{C}$$ at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$. Let $$\mathrm{MN}$$ be a chord of $$\mathrm{C}$$ of length 2 unit and slope $$-1$$. Then, a distance (in units) between... | [{"identifier": "A", "content": "$$3-\\sqrt{2}$$\n"}, {"identifier": "B", "content": "$$2-\\sqrt{3}$$\n"}, {"identifier": "C", "content": "$$\\sqrt{2}-1$$\n"}, {"identifier": "D", "content": "$$\\sqrt{2}+1$$"}] | ["A"] | null | <p>$$\text { Let the line by } x+y=\lambda$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwhfpfz5/cd7503a4-2670-4a67-8200-c99b5e7fc89f/49a14910-1803-11ef-b156-f754785ad3ce/file-1lwhfpfz6.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lwhfpfz5/cd7503a4-2670-4a67-8200-... | mcq | jee-main-2024-online-4th-april-evening-shift | 5,246 |
R6msSj4yWbFMiKDF | maths | circle | family-of-circle | A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is : | [{"identifier": "A", "content": "an ellipse"}, {"identifier": "B", "content": "a circle "}, {"identifier": "C", "content": "a hyperbola "}, {"identifier": "D", "content": "a parabola "}] | ["D"] | null | Equation of circle with center $$(0,3)$$ and radius $$2$$ is
<br><br>$${x^2} + {\left( {y - 3} \right)^2} = 4$$
<br><br>Let locus of the variable circle is $$\left( {\alpha ,\beta } \right)$$
<br><br>As it touches $$x$$-axis.
<br><br>$$\therefore$$ It's equation is $${\left( {x - \alpha } \right)^2} + {\left( {y + \... | mcq | aieee-2005 | 5,250 |
uxc61AnvgIDOB2bO | maths | circle | family-of-circle | Consider a family of circles which are passing through the point $$(-1, 1)$$ and are tangent to $$x$$-axis. If $$(h, k)$$ are the coordinate of the centre of the circles, then the set of values of $$k$$ is given by the interval : | [{"identifier": "A", "content": "$$ - {1 \\over 2} \\le k \\le {1 \\over 2}$$ "}, {"identifier": "B", "content": "$$k \\le {1 \\over 2}$$ "}, {"identifier": "C", "content": "$$0 \\le k \\le {1 \\over 2}$$ "}, {"identifier": "D", "content": "$$k \\ge {1 \\over 2}$$ "}] | ["D"] | null | Equation of circle whose center is $$\left( {h,k} \right)$$
<br><br>i.e $${\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {k^2}$$
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264751/exam_images/dhlchwnp8kfgccfwlmlr.webp" loading="lazy" alt="AIEEE 2007 Mathema... | mcq | aieee-2007 | 5,251 |
W5qdG7lPRwsgI7vI | maths | circle | family-of-circle | The differential equation of the family of circles with fixed radius $$5$$ units and centre on the line $$y = 2$$ is : | [{"identifier": "A", "content": "$$\\left( {x - 2} \\right){y^2} = 25 - {\\left( {y - 2} \\right)^2}$$ "}, {"identifier": "B", "content": "$$\\left( {y - 2} \\right){y^2} = 25 - {\\left( {y - 2} \\right)^2}$$"}, {"identifier": "C", "content": "$${\\left( {y - 2} \\right)^2}{y^2} = 25 - {\\left( {y - 2} \\right)^2}$$ "}... | ["C"] | null | Let the center of the circle be $$(h, 2)$$
<br><br>$$\therefore$$ Equation of circle is
<br><br>$${\left( {x - h} \right)^2} + \left( {y - 2} \right){}^2 = 25\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$
<br><br>Differentiating with respect to $$x,$$ we get
<br><br>$$2\left( {x - h} \right) + 2\left( {y - 2} \right){{dy} \... | mcq | aieee-2008 | 5,252 |
DtjBungYu7lolDmmqPBQA | maths | circle | family-of-circle | A circle passes through the points (2, 3) and (4, 5). If its centre lies on the line, $$y - 4x + 3 = 0,$$ then its radius is equal to : | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "$$\\sqrt 5 $$"}, {"identifier": "C", "content": "$$\\sqrt 2 $$"}, {"identifier": "D", "content": "1"}] | ["A"] | null | Centre $(\alpha, \beta)$ lies on line $y-4 x+3=0$
<br/><br/>Then, $\quad \beta=4 \alpha-3$
<br/><br/>And
<br/><br/>$$
\text { Radius }=\sqrt{(\alpha-2)^2+(\beta-3)^2}=\sqrt{(\alpha-4)^2+(\beta-5)^2}
$$
<br/><br/>$$
\begin{aligned}
& \alpha^2+\beta^2+ 13-4 \alpha-6 \beta=\alpha^2+\beta^2+41-8 \alpha-10 \beta \\\\
& 4 \a... | mcq | jee-main-2018-online-15th-april-morning-slot | 5,255 |
GhHPQJZMreFqETJ1UGuYQ | maths | circle | family-of-circle | Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis as a common tangent, then : | [{"identifier": "A", "content": "a, b, c are in A.P."}, {"identifier": "B", "content": "$$\\sqrt a ,\\sqrt b ,\\sqrt c $$ are in A.P"}, {"identifier": "C", "content": "$${1 \\over {\\sqrt b }} + {1 \\over {\\sqrt c }}$$ = $${1 \\over {\\sqrt a }}$$"}, {"identifier": "D", "content": "$${1 \\over {\\sqrt b }} = {1 \\ove... | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264861/exam_images/znw3pusc82f7kl1xv763.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 9th January Morning Slot Mathematics - Circle Question 121 English Explanation">
<br><br>AB = AC... | mcq | jee-main-2019-online-9th-january-morning-slot | 5,256 |
KCwGH1e7nkaCmpz92Q3rsa0w2w9jx1ytgcp | maths | circle | family-of-circle | The locus of the centres of the circles, which touch the circle, x<sup>2</sup>
+ y<sup>2</sup>
= 1 externally, also touch the y-axis and
lie in the first quadrant, is : | [{"identifier": "A", "content": "$$x = \\sqrt {1 + 2y} ,y \\ge 0$$"}, {"identifier": "B", "content": "$$y = \\sqrt {1 + 2x} ,x \\ge 0$$"}, {"identifier": "C", "content": "$$y = \\sqrt {1 + 4x} ,x \\ge 0$$"}, {"identifier": "D", "content": "$$x = \\sqrt {1 + 4y} ,y \\ge 0$$"}] | ["B"] | null | Let the centre is (h, k) & radius is h (h, k > 0)<br><br>
OP = h + 1<br><br>
<img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266464/exam_images/pevci3b1ygpkppdlpbgl.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 10th April E... | mcq | jee-main-2019-online-10th-april-evening-slot | 5,257 |
aLcjY2vZwkdplfoJYSjgy2xukfakl6kz | maths | circle | family-of-circle | The circle passing through the intersection of the circles, <br/>x<sup>2</sup> + y<sup>2</sup> – 6x = 0 and x<sup>2</sup> + y<sup>2</sup> – 4y = 0, having its centre on<br/> the line, 2x – 3y + 12 = 0, also passes through the point :
| [{"identifier": "A", "content": "(\u20133, 1)"}, {"identifier": "B", "content": "(1, \u20133) "}, {"identifier": "C", "content": "(\u20131, 3) "}, {"identifier": "D", "content": "(\u20133, 6) "}] | ["D"] | null | Let S be the circle passing through point of intersection of S<sub>1</sub> & S<sub>2</sub><br><br>$$ \therefore $$ S = S<sub>1</sub> + $$\lambda $$S<sub>2</sub> = 0<br><br>$$ \Rightarrow $$ $$S:({x^2} + {y^2} - 6x) + \lambda ({x^2} + {y^2} - 4y) = 0$$<br><br>$$ \Rightarrow $$ $$S:{x^2} + {y^2} - \left( {{6 \over {1... | mcq | jee-main-2020-online-4th-september-evening-slot | 5,259 |
NDpPPI6KPUzk1c1oIm1kmm35609 | maths | circle | family-of-circle | Let S<sub>1</sub> : x<sup>2</sup> + y<sup>2</sup> = 9 and S<sub>2</sub> : (x $$-$$ 2)<sup>2</sup> + y<sup>2</sup> = 1. Then the locus of center of a variable circle S which touches S<sub>1</sub> internally and S<sub>2</sub> externally always passes through the points : | [{"identifier": "A", "content": "$$\\left( {{1 \\over 2}, \\pm {{\\sqrt 5 } \\over 2}} \\right)$$"}, {"identifier": "B", "content": "(1, $$\\pm$$ 2)"}, {"identifier": "C", "content": "$$\\left( {2, \\pm {3 \\over 2}} \\right)$$"}, {"identifier": "D", "content": "(0, $$\\pm$$ $$\\sqrt 3 $$)"}] | ["C"] | null | S<sub>1</sub> : x<sup>2</sup> + y<sup>2</sup> = 9 ; C<sub>1</sub> (0, 0), r<sub>1</sub> = 3<br><br>S<sub>2</sub> : (x $$-$$ 2)<sup>2</sup> + y<sup>2</sup> = 1 ; C<sub>2</sub> (2, 0), r<sub>2</sub> = 1<br><br>Image<br><br>Let the variable circle S and its radius is r units.<br><br>Here S and S<sub>1</sub> touches intern... | mcq | jee-main-2021-online-18th-march-evening-shift | 5,260 |
1krrorbwe | maths | circle | family-of-circle | Let r<sub>1</sub> and r<sub>2</sub> be the radii of the largest and smallest circles, respectively, which pass through the point ($$-$$4, 1) and having their centres on the circumference of the circle x<sup>2</sup> + y<sup>2</sup> + 2x + 4y $$-$$ 4 = 0. If $${{{r_1}} \over {{r_2}}} = a + b\sqrt 2 $$, then a + b is equa... | [{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "11"}, {"identifier": "C", "content": "5"}, {"identifier": "D", "content": "7"}] | ["C"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264563/exam_images/bn5acdhhem4hvhtrnsbo.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265467/exam_images/qc4gbepselvkks5k9sik.webp"><source media="(max-wid... | mcq | jee-main-2021-online-20th-july-evening-shift | 5,261 |
1kteov0d4 | maths | circle | family-of-circle | Let the equation x<sup>2</sup> + y<sup>2</sup> + px + (1 $$-$$ p)y + 5 = 0 represent circles of varying radius r $$\in$$ (0, 5]. Then the number of elements in the set S = {q : q = p<sup>2</sup> and q is an integer} is __________. | [] | null | 61 | $$r = \sqrt {{{{p^2}} \over 4} + {{{{(1 - p)}^2}} \over 4} - 5} = {{\sqrt {2{p^2} - 2p - 19} } \over 2}$$<br><br>Since, $$r \in (0,5]$$<br><br>So, $$0 < 2{p^2} - 2p - 19 \le 100$$<br><br>$$ \Rightarrow p \in \left[ {{{1 - \sqrt {239} } \over 2},{{1 - \sqrt {39} } \over 2}} \right) \cup \left( {{{1 + \sqrt {39} } \o... | integer | jee-main-2021-online-27th-august-morning-shift | 5,262 |
1l589r0yu | maths | circle | family-of-circle | <p>Let C be a circle passing through the points A(2, $$-$$1) and B(3, 4). The line segment AB s not a diameter of C. If r is the radius of C and its centre lies on the circle $${(x - 5)^2} + {(y - 1)^2} = {{13} \over 2}$$, then r<sup>2</sup> is equal to :</p> | [{"identifier": "A", "content": "32"}, {"identifier": "B", "content": "$${{65} \\over 2}$$"}, {"identifier": "C", "content": "$${{61} \\over 2}$$"}, {"identifier": "D", "content": "30"}] | ["B"] | null | <p>Equation of perpendicular bisector of AB is</p>
<p>$$y - {3 \over 2} = - {1 \over 5}\left( {x - {5 \over 2}} \right) \Rightarrow x + 5y = 10$$</p>
<p>Solving it with equation of given circle,</p>
<p>$${(x - 5)^2}{\left( {{{10 - x} \over 5} - 1} \right)^2} = {{13} \over 2}$$</p>
<p>$$ \Rightarrow {(x - 5)^2}\left( {... | mcq | jee-main-2022-online-26th-june-morning-shift | 5,263 |
1ldpsfpye | maths | circle | family-of-circle | <p>Let a circle $$C_{1}$$ be obtained on rolling the circle $$x^{2}+y^{2}-4 x-6 y+11=0$$ upwards 4 units on the tangent $$\mathrm{T}$$ to it at the point $$(3,2)$$. Let $$C_{2}$$ be the image of $$C_{1}$$ in $$\mathrm{T}$$. Let $$A$$ and $$B$$ be the centers of circles $$C_{1}$$ and $$C_{2}$$ respectively, and $$M$$ an... | [{"identifier": "A", "content": "$$2\\left( {2 + \\sqrt 2 } \\right)$$"}, {"identifier": "B", "content": "$$4\\left( {1 + \\sqrt 2 } \\right)$$"}, {"identifier": "C", "content": "$$3 + 2\\sqrt 2 $$"}, {"identifier": "D", "content": "$$2\\left( {1 + \\sqrt 2 } \\right)$$"}] | ["B"] | null | Given circle is $x^{2}+y^{2}-4 x-6 y+11=0$, centre $(2,3)$
<br/><br/>Tangent at $(3,2)$ is $x-y=1$
<br/><br/>After rolling up by 4 units centre of $C_{1}$ is
<br/><br/>$A \equiv\left(2+\frac{4}{\sqrt{2}}, 3+\frac{4}{\sqrt{2}}\right)$
<br/><br/>$\Rightarrow A=(2+2 \sqrt{2}, 3+2 \sqrt{2})$
<br/><br/>$B$ is the image... | mcq | jee-main-2023-online-31st-january-morning-shift | 5,264 |
6ss2xEl1EvF1sipa | maths | circle | intercepts-of-a-circle | The circle $${x^2} + {y^2} = 4x + 8y + 5$$ intersects the line $$3x - 4y = m$$ at two distinct points if : | [{"identifier": "A", "content": "$$ - 35 < m < 15$$ "}, {"identifier": "B", "content": "$$ 15 < m < 65$$"}, {"identifier": "C", "content": "$$ 35 < m < 85$$"}, {"identifier": "D", "content": "$$ - 85 < m < -35$$"}] | ["A"] | null | Circle $${x^2} + {y^2} - 4x - 8y - 5 = 0$$
<br><br>Center $$=(2,4),$$ Radius $$ = \sqrt {4 + 16 + 5} = 5$$
<br><br>If circle is intersecting line $$3x-4y=m,$$ at two distinct points.
<br><br>$$ \Rightarrow $$ length of perpendicular from center to the line $$ < $$ radius
<br><br>$$ \Rightarrow {{\left| {6 - 16 -... | mcq | aieee-2010 | 5,265 |
0D5SbWICmfbrSiS6nhSKX | maths | circle | intercepts-of-a-circle | If a circle C, whose radius is 3, touches externally the circle,
<br/>$${x^2} + {y^2} + 2x - 4y - 4 = 0$$ at the point (2, 2), then the length of the intercept cut by this circle C, on the x-axis is equal to : | [{"identifier": "A", "content": "$$2\\sqrt 5 $$ "}, {"identifier": "B", "content": "$$3\\sqrt 2 $$"}, {"identifier": "C", "content": "$$\\sqrt 5 $$"}, {"identifier": "D", "content": "$$2\\sqrt 3 $$"}] | ["A"] | null | Given circle is :
<br><br>x<sup>2</sup> + y<sup>2</sup> + 2x $$-$$ 4y $$-$$4 = 0
<br><br>$$\therefore\,\,\,$$ its center is ($$-$$ 1, 2) and radius is 3 units.
<br><br>Let A = (x, y) be the center of the circle C
<br><br>$$ \therefore $$$$\,\,\,$$ $${{x - 1} \over 2}$$ = 2 $$ \Rightarrow $$ x = 5 and $${{y + 2} \over ... | mcq | jee-main-2018-online-16th-april-morning-slot | 5,266 |
dGyz2Ms65BQBVjVOx30tZ | maths | circle | intercepts-of-a-circle | A square is inscribed in the circle x<sup>2</sup> + y<sup>2</sup>
– 6x + 8y – 103 = 0 with its sides parallel to the coordinate axes.
Then the distance of the vertex of this square which is nearest to the origin is : | [{"identifier": "A", "content": "$$\\sqrt {137} $$"}, {"identifier": "B", "content": "6"}, {"identifier": "C", "content": "$$\\sqrt {41} $$"}, {"identifier": "D", "content": "13"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266267/exam_images/raqnosna6rdmei2xv6vs.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 11th January Morning Slot Mathematics - Circle Question 116 English Explanation">
<br>R $$ = \sq... | mcq | jee-main-2019-online-11th-january-morning-slot | 5,268 |
MNZ8HDiEA2opWMQFzoaec | maths | circle | intercepts-of-a-circle | If a circle of radius R passes through the origin O and intersects the coordinates axes at A and B, then the
locus of the foot of perpendicular from O on AB is : | [{"identifier": "A", "content": "(x<sup>2</sup> + y<sup>2</sup>)<sup>2</sup> = 4R<sup>2</sup>x<sup>2</sup>y<sup>2</sup>"}, {"identifier": "B", "content": "(x<sup>2</sup> + y<sup>2</sup>) (x + y) = R<sup>2</sup>xy"}, {"identifier": "C", "content": "(x<sup>2</sup> + y<sup>2</sup>)<sup>2</sup> = 4R<sup></sup>x<sup>2</sup>... | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266292/exam_images/mqtptxaki1efkgjktaz3.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 12th January Evening Slot Mathematics - Circle Question 112 English Explanation">
<br>Slope of A... | mcq | jee-main-2019-online-12th-january-evening-slot | 5,269 |
zzLYX99HzUqacuDxIm3rsa0w2w9jxaykypb | maths | circle | intercepts-of-a-circle | A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the
point : | [{"identifier": "A", "content": "(1, 5)"}, {"identifier": "B", "content": "( 2, 3)"}, {"identifier": "C", "content": "(3, 5)"}, {"identifier": "D", "content": "(3, 10)"}] | ["D"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265069/exam_images/iuwti6tm5cysz6ebxdbi.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 12th April Evening Slot Mathematics - Circle Question 102 English Explanation"><br><br>
From the... | mcq | jee-main-2019-online-12th-april-evening-slot | 5,271 |
ZkMgZkefk4fu6UwEbm1kmixaf77 | maths | circle | intercepts-of-a-circle | Let the lengths of intercepts on x-axis and y-axis made by the circle <br/>x<sup>2</sup> + y<sup>2</sup> + ax + 2ay + c = 0, (a < 0) be 2$${\sqrt 2 }$$ and 2$${\sqrt 5 }$$, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line x + 2y = 0, is equal to : | [{"identifier": "A", "content": "$${\\sqrt {10} }$$"}, {"identifier": "B", "content": "$${\\sqrt {6} }$$"}, {"identifier": "C", "content": "$${\\sqrt {11} }$$"}, {"identifier": "D", "content": "$${\\sqrt {7} }$$"}] | ["B"] | null | $$2\sqrt {{{{a^2}} \over 4} - c} = 2\sqrt 2 $$<br><br>$$\sqrt {{a^2} - 4c} = 2\sqrt 2 $$<br><br>$${a^2} - 4c = 8$$ .... (1)<br><br>$$2\sqrt {{a^2} - c} = 2\sqrt 5 $$<br><br>$${a^2} - c = 5$$ .... (2)<br><br>$$(2) - (1)$$<br><br>$$3c = - 3a \Rightarrow c = - 1$$<br><br>$${a^2} = 4 \Rightarrow a = - 2$$ (Given a &... | mcq | jee-main-2021-online-16th-march-evening-shift | 5,272 |
1kryesdeh | maths | circle | intercepts-of-a-circle | Consider a circle C which touches the y-axis at (0, 6) and cuts off an intercept $$6\sqrt 5 $$ on the x-axis. Then the radius of the circle C is equal to : | [{"identifier": "A", "content": "$$\\sqrt {53} $$"}, {"identifier": "B", "content": "9"}, {"identifier": "C", "content": "8"}, {"identifier": "D", "content": "$$\\sqrt {82} $$"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266278/exam_images/kb34kqlzzf1i40yfp522.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 27th July Evening Shift Mathematics - Circle Question 75 English Explanation"><br><br>$$r = \sqrt ... | mcq | jee-main-2021-online-27th-july-evening-shift | 5,273 |
1l6jdc4r8 | maths | circle | intercepts-of-a-circle | <p>If the circle $$x^{2}+y^{2}-2 g x+6 y-19 c=0, g, c \in \mathbb{R}$$ passes through the point $$(6,1)$$ and its centre lies on the line $$x-2 c y=8$$, then the length of intercept made by the circle on $$x$$-axis is :</p> | [{"identifier": "A", "content": "$$\\sqrt{11}$$"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "$$2 \\sqrt{23}$$"}] | ["D"] | null | <p>Circle : $${x^2} + {y^2} - 2gx + 6y - 19c = 0$$</p>
<p>It passes through $$h(6,1)$$</p>
<p>$$ \Rightarrow 36 + 1 - 12g + 6 - 19c = 0$$</p>
<p>$$ = 12g + 19c = 43$$ ..... (1)</p>
<p>Line $$x - 2cy = 8$$ passes through centre</p>
<p>$$ \Rightarrow g + 6c = 8$$ ...... (2)</p>
<p>From (1) & (2)</p>
<p>$$g = 2,\,c = 1$$<... | mcq | jee-main-2022-online-27th-july-morning-shift | 5,274 |
1lgrgje37 | maths | circle | intercepts-of-a-circle | <p>Two circles in the first quadrant of radii $$r_{1}$$ and $$r_{2}$$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $$x+y=2$$. Then $$r_{1}^{2}+r_{2}^{2}-r_{1} r_{2}$$ is equal to ___________.</p> | [] | null | 7 | $$
\begin{aligned}
& \text { Circle }(x-a)^2+(y-a)^2=a^2 \\\\
& x^2+y^2-2 a x-2 a y+a^2=0 \\\\
& \text { intercept }=2 \\\\
& \Rightarrow 2 \sqrt{a^2-d^2}=2
\end{aligned}
$$
<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1li9wqjod/6e47a7f9-a970-48fa-94e5-b8b0e7d86b93/96fe4cd0-fe... | integer | jee-main-2023-online-12th-april-morning-shift | 5,275 |
1lgvq76au | maths | circle | intercepts-of-a-circle | <p>Let A be the point $$(1,2)$$ and B be any point on the curve $$x^{2}+y^{2}=16$$. If the centre of the locus of the point P, which divides the line segment $$\mathrm{AB}$$ in the ratio $$3: 2$$ is the point C$$(\alpha, \beta)$$, then the length of the line segment $$\mathrm{AC}$$ is :</p> | [{"identifier": "A", "content": "$$\\frac{3 \\sqrt{5}}{5}$$"}, {"identifier": "B", "content": "$$\\frac{6 \\sqrt{5}}{5}$$"}, {"identifier": "C", "content": "$$\\frac{2 \\sqrt{5}}{5}$$"}, {"identifier": "D", "content": "$$\\frac{4 \\sqrt{5}}{5}$$"}] | ["A"] | null | We have, equation of circle is $x^2+y^2=16$
<br><br>Let any point on the circle $x^2+y^2=4^2$ is $B(4 \cos \theta, 4 \sin \theta)$ and $A(1,2)$
<br><br>Let $\mathrm{P}$ be $(h, k)$ which divides $\mathrm{AB}$ in $3: 2$
<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lnk5g4ko/0e3a3eb2-16a6... | mcq | jee-main-2023-online-10th-april-evening-shift | 5,276 |
1lh23y6we | maths | circle | intercepts-of-a-circle | <p> A circle passing through the point $$P(\alpha, \beta)$$ in the first quadrant touches the two coordinate axes at the points $$A$$ and $$B$$. The point $$P$$ is above the line $$A B$$. The point $$Q$$ on the line segment $$A B$$ is the foot of perpendicular from $$P$$ on $$A B$$. If $$P Q$$ is equal to 11 units, the... | [] | null | 121 | Let equation of circle is $(x-a)^2+(y-a)^2=a^2$
<br><br>Since, (i) passes through $P(\alpha, \beta)$
<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lo4iz7t1/8abb90cb-5a21-43f7-858e-e4e5d897d907/29530750-7288-11ee-955e-53a531d428f9/file-6y3zli1lo4iz7t2.png?format=png" data-orsrc="https:/... | integer | jee-main-2023-online-6th-april-morning-shift | 5,277 |
jaoe38c1lsd4sm2c | maths | circle | intercepts-of-a-circle | <p>Let a variable line passing through the centre of the circle $$x^2+y^2-16 x-4 y=0$$, meet the positive co-ordinate axes at the points $$A$$ and $$B$$. Then the minimum value of $$O A+O B$$, where $$O$$ is the origin, is equal to</p> | [{"identifier": "A", "content": "12"}, {"identifier": "B", "content": "20"}, {"identifier": "C", "content": "24"}, {"identifier": "D", "content": "18"}] | ["D"] | null | <p>$$\begin{aligned}
& (y-2)=m(x-8) \\
& \Rightarrow x \text {-intercept } \\
& \Rightarrow\left(\frac{-2}{m}+8\right) \\
& \Rightarrow y \text {-intercept } \\
& \Rightarrow(-8 \mathrm{~m}+2) \\
& \Rightarrow \mathrm{OA}+\mathrm{OB}=\frac{-2}{\mathrm{~m}}+8-8 \mathrm{~m}+2 \\
& \mathrm{f}^{\prime}(\mathrm{m})=\frac{2}... | mcq | jee-main-2024-online-31st-january-evening-shift | 5,278 |
pRzLsQDdQrCvWwSH | maths | circle | number-of-common-tangents-and-position-of-two-circle | The two circles x<sup>2</sup> + y<sup>2</sup> = ax, and x<sup>2</sup> + y<sup>2</sup> = c<sup>2</sup> (c > 0) touch each other if : | [{"identifier": "A", "content": "| a | = c"}, {"identifier": "B", "content": "a = 2c"}, {"identifier": "C", "content": "| a | = 2c"}, {"identifier": "D", "content": "2 | a | = c"}] | ["A"] | null | As center of one circle is $$\left( {0,0} \right)$$ and other circle passes through $$(0,0),$$ therefore
<br><br>Also $${C_1}\left( {{a \over 2},0} \right){C_2}\left( {0,0} \right)$$
<br><br>$${r_1} = {a \over 2}{r_2} = C$$
<br><br>$${C_1}{C_2} = {r_1} - {r_2} = {a \over 2}$$
<br><br>$$ \Rightarrow C - {a \over 2} = {... | mcq | aieee-2011 | 5,280 |
Dym616D1FHZIYakE | maths | circle | number-of-common-tangents-and-position-of-two-circle | Let $$C$$ be the circle with centre at $$(1, 1)$$ and radius $$=$$ $$1$$. If $$T$$ is the circle centred at $$(0, y)$$, passing through origin and touching the circle $$C$$ externally, then the radius of $$T$$ is equal to : | [{"identifier": "A", "content": "$${1 \\over 2}$$ "}, {"identifier": "B", "content": "$${1 \\over 4}$$"}, {"identifier": "C", "content": "$${{\\sqrt 3 } \\over {\\sqrt 2 }}$$ "}, {"identifier": "D", "content": "$${{\\sqrt 3 } \\over 2}$$ "}] | ["B"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263502/exam_images/az5z8h8zms4j96efvsxd.webp" loading="lazy" alt="JEE Main 2014 (Offline) Mathematics - Circle Question 137 English Explanation">
<br><br>Equation of circle $$C \equiv {\left( {x - 1} \right)^2} + {\left( {y - 1} \ri... | mcq | jee-main-2014-offline | 5,281 |
0uEqnZaNFKvdeUZb | maths | circle | number-of-common-tangents-and-position-of-two-circle | The number of common tangents to the circles $${x^2} + {y^2} - 4x - 6x - 12 = 0$$ and $${x^2} + {y^2} + 6x + 18y + 26 = 0,$$ is : | [{"identifier": "A", "content": "$$3$$"}, {"identifier": "B", "content": "$$4$$"}, {"identifier": "C", "content": "$$1$$"}, {"identifier": "D", "content": "$$2$$"}] | ["A"] | null | $${x^2} + {y^2} - 4x - 6y - 12 = 0\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$
<br><br>Center, $${c_1} = \left( {2,\,3} \right)$$ and Radius, $${r_1} = 5$$ units
<br><br>$${x^2} + {y^2} + 6x + 18y + 26 = 0\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$$
<br><br>Center, $${c_2} = \left( { - 3, - 9} \right)$$ and Radius... | mcq | jee-main-2015-offline | 5,282 |
p8FE11vq7emHA0SgNw18hoxe66ijvwub61o | maths | circle | number-of-common-tangents-and-position-of-two-circle | The common tangent to the circles x <sup>2</sup> + y<sup>2</sup> = 4 and
x<sup>2</sup> + y<sup>2</sup> + 6x + 8y – 24 = 0 also passes through the
point : | [{"identifier": "A", "content": "(6, \u20132)"}, {"identifier": "B", "content": "(4, \u20132)"}, {"identifier": "C", "content": "(\u20134, 6)"}, {"identifier": "D", "content": "(\u20136, 4)"}] | ["A"] | null | For this circle
x <sup>2</sup> + y<sup>2</sup> = 4
<br><br>Center C<sub>1</sub> = (0, 0) and radius r<sub>1</sub> = 2
<br><br>For this circle
x<sup>2</sup> + y<sup>2</sup> + 6x + 8y – 24 = 0
<br><br>Center C<sub>2</sub> = (-3, -4) and radius r<sub>2</sub> = $$\sqrt {9 + 16 + 24} $$ = 7
<br><br>So distance between cente... | mcq | jee-main-2019-online-9th-april-evening-slot | 5,284 |
RQXU1QZmaI8cAm09q9pZQ | maths | circle | number-of-common-tangents-and-position-of-two-circle | Let C<sub>1</sub> and C<sub>2</sub> be the centres of the circles x<sup>2</sup> + y<sup>2</sup> – 2x – 2y – 2 = 0 and x<sup>2</sup> + y<sup>2</sup> – 6x – 6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral PC<sub>1</sub>QC<sub>2</sub>... | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "6"}, {"identifier": "C", "content": "9"}, {"identifier": "D", "content": "8"}] | ["A"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263648/exam_images/kjqkelq5u313u7up3q3r.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 12th January Morning Slot Mathematics - Circle Question 113 English Explanation">
<br>Area = 2 ... | mcq | jee-main-2019-online-12th-january-morning-slot | 5,285 |
spzMVkSBmK5b0Xg5TlTdc | maths | circle | number-of-common-tangents-and-position-of-two-circle | If a variable line, 3x + 4y – $$\lambda $$ = 0 is such that the two circles x<sup>2</sup> + y<sup>2</sup> – 2x – 2y + 1 = 0 and x<sup>2</sup> + y<sup>2</sup> – 18x – 2y + 78 = 0 are on its opposite sides, then the set of all values of $$\lambda $$ is the interval : | [{"identifier": "A", "content": "(23, 31)"}, {"identifier": "B", "content": "(2, 17)"}, {"identifier": "C", "content": "[13, 23] "}, {"identifier": "D", "content": "[12, 21] "}] | ["D"] | null | Centre of circles are opposite side of line
<br><br>(3 + 4 $$-$$ $$\lambda $$) (27 + 4 $$-$$ $$\lambda $$) < 0
<br><br>($$\lambda $$ $$-$$ 7) ($$\lambda $$ $$-$$ 31) < 0
<br><br>$$\lambda $$ $$ \in $$ (7, 31)
<br><br>distance from S<sub>1</sub>
<br><br>$$\left| {{{3 + 4 - \lambda } \over 5}} \right| \ge 1 \Righta... | mcq | jee-main-2019-online-12th-january-morning-slot | 5,286 |
zbBZfrh5YlxIoiF5Ce8if | maths | circle | number-of-common-tangents-and-position-of-two-circle | Two circles with equal radii are intersecting at the points (0, 1) and (0, –1). The tangent at the point (0, 1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is : | [{"identifier": "A", "content": "$$2\\sqrt 2 $$"}, {"identifier": "B", "content": "$$\\sqrt 2 $$"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "1"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263285/exam_images/yhmndiwp2ujk1aofixgb.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 11th January Morning Slot Mathematics - Circle Question 115 English Explanation">
<br><br>In $$\... | mcq | jee-main-2019-online-11th-january-morning-slot | 5,287 |
BRTMkdqUdllpf8r6uf8jG | maths | circle | number-of-common-tangents-and-position-of-two-circle | If the circles
<br/><br/>x<sup>2</sup> + y<sup>2</sup> $$-$$ 16x $$-$$ 20y + 164 = r<sup>2</sup>
<br/><br/>and (x $$-$$ 4)<sup>2</sup> + (y $$-$$ 7)<sup>2</sup> = 36
<br/><br/>intersect at two distinct points, then : | [{"identifier": "A", "content": "r > 11"}, {"identifier": "B", "content": "0 < r < 1"}, {"identifier": "C", "content": "r = 11"}, {"identifier": "D", "content": "1 < r < 11"}] | ["D"] | null | Circles are x<sup>2</sup> + y<sup>2</sup> $$-$$ 16x $$-$$ 20y + 164 = r<sup>2</sup> $$ \Rightarrow $$ c<sub>1</sub> (8, 10)
<br><br>and (x $$-$$ 4)<sup>2</sup> + (y $$-$$ 7)<sup>2</sup> = 36
<br><br>they intersect at two distinct points
<br><br>$$\left| {{r_1} - {r_2}} \right| < {c_1}{c_2} < {r_1} + {r_2}\le... | mcq | jee-main-2019-online-9th-january-evening-slot | 5,288 |
9GoPmJBIeKYfTPS79ujgy2xukewts732 | maths | circle | number-of-common-tangents-and-position-of-two-circle | The number of integral values of k for which
the line, 3x + 4y = k intersects the circle,
<br/>x<sup>2</sup>
+ y<sup>2</sup>
– 2x – 4y + 4 = 0 at two distinct points is
______. | [] | null | 9 | Circle x<sup>2</sup>
+ y<sup>2</sup>
– 2x – 4y + 4 = 0
<br><br>$$ \Rightarrow $$ (x – 1)<sup>2</sup>
+ (y – 2)<sup>2</sup>
= 1
<br><br>Centre: (1, 2), radius = 1
<br><br>Line 3x + 4y – k = 0 intersects the circle at two distinct points.
<br><br>$$ \Rightarrow $$ distance of centre from the line < radius
<br><br>$$... | integer | jee-main-2020-online-2nd-september-morning-slot | 5,289 |
ppjkwHdAOgYLtIrc8d1kmjb64v2 | maths | circle | number-of-common-tangents-and-position-of-two-circle | Choose the incorrect statement about the two circles whose equations are given below :<br/><br/>x<sup>2</sup> + y<sup>2</sup> $$-$$ 10x $$-$$ 10y + 41 = 0 and <br/><br/>x<sup>2</sup> + y<sup>2</sup> $$-$$ 16x $$-$$ 10y + 80 = 0 | [{"identifier": "A", "content": "Distance between two centres is the average of radii of both the circles."}, {"identifier": "B", "content": "Both circles pass through the centre of each other."}, {"identifier": "C", "content": "Circles have two intersection points."}, {"identifier": "D", "content": "Both circle's cent... | ["D"] | null | S<sub>1</sub> $$ \equiv $$ x<sup>2</sup> + y<sup>2</sup> $$-$$ 10x $$-$$ 10y + 41 = 0<br><br>Centre C<sub>1</sub> $$ \equiv $$ (5, 5), radius r<sub>1</sub> = 3<br><br>S<sub>2</sub> $$ \equiv $$ x<sup>2</sup> + y<sup>2</sup> $$-$$ 16x $$-$$ 10y + 80 = 0<br><br>Centre C<sub>2</sub> $$ \equiv $$ (8, 5), radius r<sub>2</su... | mcq | jee-main-2021-online-17th-march-morning-shift | 5,290 |
jj6PYUtpBmn4DW8BTT1kmjco2b6 | maths | circle | number-of-common-tangents-and-position-of-two-circle | The minimum distance between any two points P<sub>1</sub> and P<sub>2</sub> while considering point P<sub>1</sub> on one circle and point P<sub>2</sub> on the other circle for the given circles' equations<br/><br/>x<sup>2</sup> + y<sup>2</sup> $$-$$ 10x $$-$$ 10y + 41 = 0<br/><br/>x<sup>2</sup> + y<sup>2</sup> $$-$$ 24... | [] | null | 1 | $${S_1}:{(x - 5)^2} + {(y - 5)^2} = 9$$
<br><br>Centre (5, 5), r<sub>1</sub> = 3<br><br>$${S_2}:{(x - 12)^2} + {(y - 5)^2} = 9$$
<br><br>Centre (12, 5), r<sub>2</sub> = 3<br><br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264982/exam_images/rpihjtwuabjqwdlwqq9s.webp" style="max-width: 100%;height... | integer | jee-main-2021-online-17th-march-morning-shift | 5,291 |
vSsFERngWYuSBpA34g1kmli5r94 | maths | circle | number-of-common-tangents-and-position-of-two-circle | Choose the correct statement about two circles whose equations are given below :<br/><br/>x<sup>2</sup> + y<sup>2</sup> $$-$$ 10x $$-$$ 10y + 41 = 0<br/><br/>x<sup>2</sup> + y<sup>2</sup> $$-$$ 22x $$-$$ 10y + 137 = 0 | [{"identifier": "A", "content": "circles have same centre"}, {"identifier": "B", "content": "circles have no meeting point"}, {"identifier": "C", "content": "circles have only one meeting point"}, {"identifier": "D", "content": "circles have two meeting points"}] | ["C"] | null | Let $${S_1}:{x^2} + {y^2} - 10x - 10y + 41 = 0$$<br><br>$$ \Rightarrow {(x - 5)^2} + {(y - 5)^2} = 9$$<br><br>Centre $$({C_1}) = (5,5)$$<br><br>Radius r<sub>1</sub> = 3<br><br>$${S_2}:{x^2} + {y^2} - 22x - 10y + 137 = 0$$<br><br>$$ \Rightarrow {(x - 11)^2} + {(y - 5)^2} = 9$$<br><br>Centre $$({C_2}) = (11,5)$$<br><br>R... | mcq | jee-main-2021-online-18th-march-morning-shift | 5,292 |
lgnx2ht3 | maths | circle | number-of-common-tangents-and-position-of-two-circle | The number of common tangents, to the circles <br/><br/>$x^{2}+y^{2}-18 x-15 y+131=0$ <br/><br/>and $x^{2}+y^{2}-6 x-6 y-7=0$, is : | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "1"}] | ["C"] | null | <p>We are given two circles:</p>
<p>(1) $x^2+y^2-18 x-15 y+131=0$</p>
<p>(2) $x^2+y^2-6 x-6 y-7=0$</p>
<p>First, let's find the centers and radii of the circles.</p>
<p>For circle (1):</p>
<p>Completing the square for the equation:</p>
<p>$(x^2-18x+{81})+(y^2-15y+\frac{225}{4})=-131+{81}+\frac{225}{4}$</p>
<p>$(x-9... | mcq | jee-main-2023-online-15th-april-morning-shift | 5,294 |
1lsg8yeao | maths | circle | number-of-common-tangents-and-position-of-two-circle | <p>If the circles $$(x+1)^2+(y+2)^2=r^2$$ and $$x^2+y^2-4 x-4 y+4=0$$ intersect at exactly two distinct points, then</p> | [{"identifier": "A", "content": "$$\\frac{1}{2}<\\mathrm{r}<7$$\n"}, {"identifier": "B", "content": "$$3<\\mathrm{r}<7$$\n"}, {"identifier": "C", "content": "$$5<\\mathrm{r}<9$$\n"}, {"identifier": "D", "content": "$$0<\\mathrm{r}<7$$"}] | ["B"] | null | <p>If two circles intersect at two distinct points</p>
<p>$$\begin{aligned}
& \Rightarrow\left|\mathrm{r}_1-\mathrm{r}_2\right|<\mathrm{C}_1 \mathrm{C}_2<\mathrm{r}_1+\mathrm{r}_2 \\
& |\mathrm{r}-2|<\sqrt{9+16}<\mathrm{r}+2 \\
& |\mathrm{r}-2|<5 \text { and } \mathrm{r}+2>5 \\
& -5<\mathrm{r}-2<5 \quad \mathrm{r}>3 ~\... | mcq | jee-main-2024-online-30th-january-morning-shift | 5,295 |
gIWxzoGle7kZLLADRm7k9k2k5kia3nk | maths | circle | orthogonality-of-two-circles | If the curves, x<sup>2</sup> – 6x + y<sup>2</sup> + 8 = 0 and
<br/>x<sup>2</sup> – 8y + y<sup>2</sup> + 16 – k = 0, (k > 0) touch each other
at a point, then the largest value of k is ______. | [] | null | 36 | C<sub>1</sub> : x<sup>2</sup> + y<sup>2</sup> – 6x + + 8 = 0
<br><br>C<sub>1</sub>(3, 0) and r<sub>1</sub> = 1
<br><br>C<sub>2</sub> : x<sup>2</sup> + y<sup>2</sup> – 8y + 16 – k = 0
<br><br>C<sub>2</sub>(0, 4) and r<sub>2</sub> = $$\sqrt k $$
<br><br>Two circles touch each other
<br><br>$$ \therefore $$ C<sub>1</sub... | integer | jee-main-2020-online-9th-january-evening-slot | 5,300 |
1l6p3thnw | maths | circle | orthogonality-of-two-circles | <p>Let the mirror image of a circle $$c_{1}: x^{2}+y^{2}-2 x-6 y+\alpha=0$$ in line $$y=x+1$$ be $$c_{2}: 5 x^{2}+5 y^{2}+10 g x+10 f y+38=0$$. If $$\mathrm{r}$$ is the radius of circle $$\mathrm{c}_{2}$$, then $$\alpha+6 \mathrm{r}^{2}$$ is equal to ________.</p> | [] | null | 12 | <p>$${c_1}:{x^2} + {y^2} - 2x - 6y + \alpha = 0$$</p>
<p>Then centre $$ = (1,3)$$ and radius $$(r) = \sqrt {10 - \alpha } $$</p>
<p>Image of $$(1,3)$$ w.r.t. line $$x - y + 1 = 0$$ is $$(2,2)$$</p>
<p>$${c_2}:5{x^2} + 5{y^2} + 10gx + 10fy + 38 = 0$$</p>
<p>or $${x^2} + {y^2} + 2gx + 2fy + {{38} \over 5} = 0$$</p>
<p>T... | integer | jee-main-2022-online-29th-july-morning-shift | 5,301 |
1ktgpx85g | maths | circle | pair-of-tangents | Two circles each of radius 5 units touch each other at the point (1, 2). If the equation of their common tangent is 4x + 3y = 10, and C<sub>1</sub>($$\alpha$$, $$\beta$$) and C<sub>2</sub>($$\gamma$$, $$\delta$$), C<sub>1</sub> $$\ne$$ C<sub>2</sub> are their centres, then |($$\alpha$$ + $$\beta$$) ($$\gamma$$ + $$\del... | [] | null | 40 | Slope of line joining centres of circles = $${4 \over 3} = \tan \theta $$<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1kx5g1hrm/3c7cc7eb-253c-4f70-8baa-06a68070a4ec/9eab2120-5c7f-11ec-bf3c-e32253ee0710/file-1kx5g1hrn.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1kx5g... | integer | jee-main-2021-online-27th-august-evening-shift | 5,303 |
1l5w01mde | maths | circle | pair-of-tangents | <p>Consider three circles:</p>
<p>$${C_1}:{x^2} + {y^2} = {r^2}$$</p>
<p>$${C_2}:{(x - 1)^2} + {(y - 1)^2} = {r^2}$$</p>
<p>$${C_3}:{(x - 2)^2} + {(y - 1)^2} = {r^2}$$</p>
<p>If a line L : y = mx + c be a common tangent to C<sub>1</sub>, C<sub>2</sub> and C<sub>3</sub> such that C<sub>1</sub> and C<sub>3</sub> lie on o... | [{"identifier": "A", "content": "23"}, {"identifier": "B", "content": "15"}, {"identifier": "C", "content": "12"}, {"identifier": "D", "content": "6"}] | ["D"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5ykyyx8/21715b9a-e296-480d-b13d-bb0f765bfb24/5c010fd0-0ae7-11ed-a51c-73986e88f75f/file-1l5ykyyx9.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5ykyyx8/21715b9a-e296-480d-b13d-bb0f765bfb24/5c010fd0-0ae7-11ed-a51c-73986e88f75f... | mcq | jee-main-2022-online-30th-june-morning-shift | 5,304 |
pEYxJxihaVS8EGPeEO1kluvq0eh | maths | circle | position-of-a-point-with-respect-to-circle | Let A(1, 4) and B(1, $$-$$5) be two points. Let P be a point on the circle <br/>(x $$-$$ 1)<sup>2</sup> + (y $$-$$ 1)<sup>2</sup> = 1 such that (PA)<sup>2</sup> + (PB)<sup>2</sup> have maximum value, then the points, P, A and B lie on : | [{"identifier": "A", "content": "a straight line"}, {"identifier": "B", "content": "an ellipse"}, {"identifier": "C", "content": "a parabola"}, {"identifier": "D", "content": "a hyperbola"}] | ["A"] | null | P be a point on $${(x - 1)^2} + {(y - 1)^2} = 1$$<br><br>so $$P(1 + \cos \theta ,1 + \sin \theta )$$<br><br>A(1, 4), B(1, $$-$$5)<br><br>$${(PA)^2} + {(PB)^2}$$<br><br>$$ = {(\cos \theta )^2} + {(\sin \theta - 3)^2} + {(\cos \theta )^2} + {(\sin \theta + 6)^2}$$<br><br>$$ = 47 + 6\sin \theta $$<br><br>It is maximum i... | mcq | jee-main-2021-online-26th-february-evening-slot | 5,307 |
yqOGL0vZWY5owc1JTs1kmliqxwh | maths | circle | position-of-a-point-with-respect-to-circle | For the four circles M, N, O and P, following four equations are given :<br/><br/>Circle M : x<sup>2</sup> + y<sup>2</sup> = 1<br/><br/>Circle N : x<sup>2</sup> + y<sup>2</sup> $$-$$ 2x = 0<br/><br/>Circle O : x<sup>2</sup> + y<sup>2</sup> $$-$$ 2x $$-$$ 2y + 1 = 0<br/><br/>Circle P : x<sup>2</sup> + y<sup>2</sup> $$-$... | [{"identifier": "A", "content": "Rhombus"}, {"identifier": "B", "content": "Square"}, {"identifier": "C", "content": "Rectangle"}, {"identifier": "D", "content": "Parallelogram"}] | ["B"] | null | $${C_M} = (0,0)$$<br><br>$${C_N} = (1,0)$$<br><br>$${C_O} = (1,1)$$<br><br>$${C_P} = (0,1)$$<br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266015/exam_images/tynzdnx9ipskg5hfjo3r.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 18... | mcq | jee-main-2021-online-18th-march-morning-shift | 5,308 |
1kru4cxvo | maths | circle | position-of-a-point-with-respect-to-circle | Let the circle S : 36x<sup>2</sup> + 36y<sup>2</sup> $$-$$ 108x + 120y + C = 0 be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, x $$-$$ 2y = 4 and 2x $$-$$ y = 5 lies inside the circle S, then : | [{"identifier": "A", "content": "$${{25} \\over 9} < C < {{13} \\over 3}$$"}, {"identifier": "B", "content": "100 < C < 165"}, {"identifier": "C", "content": "81 < C < 156"}, {"identifier": "D", "content": "100 < C < 156"}] | ["D"] | null | S : 36x<sup>2</sup> + 36y<sup>2</sup> $$-$$ 108x + 120y + C = 0<br><br>$$\Rightarrow$$ x<sup>2</sup> + y<sup>2</sup> $$-$$ 3x + $${{10} \over 3}$$y + $${C \over {36}}$$ = 0<br><br>Centre $$ \equiv ( - g, - f) \equiv \left( {{3 \over 2},{{ - 10} \over 6}} \right)$$<br><br>radius = $$r = \sqrt {{9 \over 4} + {{100} \over... | mcq | jee-main-2021-online-22th-july-evening-shift | 5,309 |
1ktislkyb | maths | circle | position-of-a-point-with-respect-to-circle | If the variable line 3x + 4y = $$\alpha$$ lies between the two <br/>circles (x $$-$$ 1)<sup>2</sup> + (y $$-$$ 1)<sup>2</sup> = 1 <br/>and (x $$-$$ 9)<sup>2</sup> + (y $$-$$ 1)<sup>2</sup> = 4, without intercepting a chord on either circle, then the sum of all the integral values of $$\alpha$$ is ___________. | [] | null | 165 | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263671/exam_images/tmz0n9h2htagrkfozwru.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 31st August Morning Shift Mathematics - Circle Question 65 English Explanation"><br><br>Both cente... | integer | jee-main-2021-online-31st-august-morning-shift | 5,310 |
1l56r83vp | maths | circle | position-of-a-point-with-respect-to-circle | <p>The set of values of k, for which the circle $$C:4{x^2} + 4{y^2} - 12x + 8y + k = 0$$ lies inside the fourth quadrant and the point $$\left( {1, - {1 \over 3}} \right)$$ lies on or inside the circle C, is :</p> | [{"identifier": "A", "content": "an empty set"}, {"identifier": "B", "content": "$$\\left( {6,{{65} \\over 9}} \\right]$$"}, {"identifier": "C", "content": "$$\\left[ {{{80} \\over 9},10} \\right)$$"}, {"identifier": "D", "content": "$$\\left( {9,{{92} \\over 9}} \\right]$$"}] | ["D"] | null | <p>$$C:4{x^2} + 4{y^2} - 12x + 8y + k = 0$$</p>
<p>$$\because$$ $$\left( {1, - {1 \over 3}} \right)$$ lies on or inside the C</p>
<p>then $$4 + {4 \over 9} - 12 - {8 \over 3} + k \le 0$$</p>
<p>$$ \Rightarrow k \le {{92} \over 9}$$</p>
<p>Now, circle lies in 4<sup>th</sup> quadrant centre $$ \equiv \left( {{3 \over 2},... | mcq | jee-main-2022-online-27th-june-evening-shift | 5,311 |
lsbkwd8l | maths | circle | position-of-a-point-with-respect-to-circle | Four distinct points $(2 k, 3 k),(1,0),(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to : | [{"identifier": "A", "content": "$\\frac{3}{13}$"}, {"identifier": "B", "content": "$\\frac{2}{13}$"}, {"identifier": "C", "content": "$\\frac{5}{13}$"}, {"identifier": "D", "content": "$\\frac{1}{13}$"}] | ["C"] | null | <p>$$(2 k, 3 k)$$ will lie on circle whose diameter is $$A B$$.</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lt1dia1n/7e33c93b-3bc8-4d95-93f9-8ca9b48aaae2/7f9962b0-d3c9-11ee-a50b-bb659a2e1d74/file-1lt1dia1o.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lt1dia1n/7e33c... | mcq | jee-main-2024-online-27th-january-morning-shift | 5,313 |
jaoe38c1lsf0k4jf | maths | circle | position-of-a-point-with-respect-to-circle | <p>Equations of two diameters of a circle are $$2 x-3 y=5$$ and $$3 x-4 y=7$$. The line joining the points $$\left(-\frac{22}{7},-4\right)$$ and $$\left(-\frac{1}{7}, 3\right)$$ intersects the circle at only one point $$P(\alpha, \beta)$$. Then, $$17 \beta-\alpha$$ is equal to _________.</p> | [] | null | 2 | <p>Centre of circle is $$(1,-1)$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lt2z9gaq/def8db7c-9d66-41d8-85bb-325814b0bd97/5ba32a20-d4ab-11ee-bdd1-01c80c3e2d9a/file-1lt2z9gar.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lt2z9gaq/def8db7c-9d66-41d8-85bb-325814b0bd9... | integer | jee-main-2024-online-29th-january-morning-shift | 5,314 |
lvb294qv | maths | circle | position-of-a-point-with-respect-to-circle | <p>If $$\mathrm{P}(6,1)$$ be the orthocentre of the triangle whose vertices are $$\mathrm{A}(5,-2), \mathrm{B}(8,3)$$ and $$\mathrm{C}(\mathrm{h}, \mathrm{k})$$, then the point $$\mathrm{C}$$ lies on the circle :</p> | [{"identifier": "A", "content": "$$x^2+y^2-74=0$$\n"}, {"identifier": "B", "content": "$$x^2+y^2-65=0$$\n"}, {"identifier": "C", "content": "$$x^2+y^2-61=0$$\n"}, {"identifier": "D", "content": "$$x^2+y^2-52=0$$"}] | ["B"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwabk4bk/5b6f7bc9-3122-44bf-ae55-efc3ca540e79/9484e510-1419-11ef-b3d9-7392e0033caf/file-1lwabk4bl.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lwabk4bk/5b6f7bc9-3122-44bf-ae55-efc3ca540e79/9484e510-1419-11ef-b3d9-7392e0033caf... | mcq | jee-main-2024-online-6th-april-evening-shift | 5,315 |
yQICdNcg13IghxZH | maths | circle | radical-axis | If the circles $${x^2}\, + \,{y^2} + \,2ax\, + \,cy\, + a\,\, = 0$$ and $${x^2}\, + \,{y^2} - \,3ax\, + \,dy\, - 1\,\, = 0$$ intersect in two ditinct points P and Q then the line 5x + by - a = 0 passes through P and Q for : | [{"identifier": "A", "content": "exactly one value of a "}, {"identifier": "B", "content": "no value of a"}, {"identifier": "C", "content": "infinitely many values of a "}, {"identifier": "D", "content": "exactly two values of a "}] | ["B"] | null | $${s_1} = {x^2} + {y^2} + 2ax + cy + a = 0$$
<br><br>$${s_2} = {x^2} + {y^2} - 3ax + dy - 1 = 0$$
<br><br>Equation of common chord of circles $${s_1}$$ and $${s_2}$$ is
<br><br>given by $${s_1} - {s_2} = 0$$
<br><br>$$ \Rightarrow 5ax + \left( {c - d} \right)y + a + 1 = 0$$
<br><br>Given that $$5x + by - a = 0$$ pas... | mcq | aieee-2005 | 5,316 |
1lsg53zah | maths | circle | radical-axis | <p>Consider two circles $$C_1: x^2+y^2=25$$ and $$C_2:(x-\alpha)^2+y^2=16$$, where $$\alpha \in(5,9)$$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $$C_1$$ and $$C_2$$ be $$\sin ^{-1}\left(\frac{\sqrt{63}}{8}\right)$$. If the length of common chord of $$C_1$$ an... | [] | null | 1575 | <p>$$\begin{gathered}
C_1: x^2+y^2=25, C_2:(x-\alpha)^2+y^2=16 \\
5<\alpha<9
\end{gathered}$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsoxip5f/ce7292b4-b5dd-4ed7-ad89-ffefc757a7b4/91687230-ccf1-11ee-a330-494dca5e9a63/file-6y3zli1lsoxip5g.png?format=png" data-orsrc="https://ap... | integer | jee-main-2024-online-30th-january-evening-shift | 5,317 |
lv9s1zqi | maths | circle | radical-axis | <p>Let the circle $$C_1: x^2+y^2-2(x+y)+1=0$$ and $$\mathrm{C_2}$$ be a circle having centre at $$(-1,0)$$ and radius 2 . If the line of the common chord of $$\mathrm{C}_1$$ and $$\mathrm{C}_2$$ intersects the $$\mathrm{y}$$-axis at the point $$\mathrm{P}$$, then the square of the distance of P from the centre of $$\ma... | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "6"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "1"}] | ["C"] | null | <p>$$\begin{gathered}
C_1: x^2+y^2-2(x+y)+1=0 \\
C_2:(x+1)^2+y^2=(2)^2 \\
x^2+y^2+2 x-3=0
\end{gathered}$$</p>
<p>Common chord is</p>
<p>$$\begin{aligned}
& C_1-C_2=0 \\
& \Rightarrow 2 x+y-2=0
\end{aligned}$$</p>
<p>also, this line intersects the $$y$$-axis at the point</p>
<p>$$\begin{aligned}
& P(y, 0) . \\
& \Right... | mcq | jee-main-2024-online-5th-april-evening-shift | 5,318 |
JSdvzTMnngQqBri2 | maths | circle | tangent-and-normal | The circle passing through $$(1, -2)$$ and touching the axis of $$x$$ at $$(3, 0)$$ also passes through the point : | [{"identifier": "A", "content": "$$\\left( { - 5,\\,2} \\right)$$ "}, {"identifier": "B", "content": "$$\\left( { 2,\\,-5} \\right)$$"}, {"identifier": "C", "content": "$$\\left( { 5,\\,-2} \\right)$$"}, {"identifier": "D", "content": "$$\\left( { - 2,\\,5} \\right)$$"}] | ["C"] | null | Since circle touches $$x$$-axis at $$(3,0)$$
<br><br>$$\therefore$$ The equation of circle be
<br><br>$${\left( {x - 3} \right)^2} + {\left( {y - 0} \right)^2} + \lambda y = 0$$
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265119/exam_images/izz5jb334xfqht2xhhp9.webp... | mcq | jee-main-2013-offline | 5,319 |
bj6fQIovF0KROgnzjJ6iB | maths | circle | tangent-and-normal | Equation of the tangent to the circle, at the point (1, −1), whose centre is the point of intersection of the straight lines x − y = 1 and 2x + y = 3 is : | [{"identifier": "A", "content": "4x + y \u2212 3 = 0"}, {"identifier": "B", "content": "x + 4y + 3 = 0"}, {"identifier": "C", "content": "3x \u2212 y \u2212 4 = 0"}, {"identifier": "D", "content": "x \u2212 3y \u2212 4 = 0"}] | ["B"] | null | Point of intersection of lines
<br><br>x $$-$$ y = 1 and 2x + y = 3 is $$\left( {{4 \over 3},{1 \over 3}} \right)$$
<br><br>Slope of OP = $${{{1 \over 3} + 1} \over {{4 \over 3} - 1}}$$ = $${{{4 \over 3}} \over {{1 \over 3}}}$$ = 4
<br><br>Slope of tangent = $$-$$ $${1 \over 4}$$
<br><br>Equatio... | mcq | jee-main-2016-online-10th-april-morning-slot | 5,320 |
4hYh8JWttCD0c1UU | maths | circle | tangent-and-normal | The radius of a circle, having minimum area, which touches the curve y = 4 – x<sup>2</sup> and the lines, y = |x| is : | [{"identifier": "A", "content": "$$2\\left( {\\sqrt 2 - 1} \\right)$$"}, {"identifier": "B", "content": "$$4\\left( {\\sqrt 2 - 1} \\right)$$"}, {"identifier": "C", "content": "$$4\\left( {\\sqrt 2 + 1} \\right)$$"}, {"identifier": "D", "content": "$$2\\left( {\\sqrt 2 + 1} \\right)$$"}] | ["B"] | null | Let the radius of circle with least area be r.
<br><br>Then, the coordinate of the center = (0, b)
<br><br>$$ \therefore $$ The equation of circle be
x<sup>2</sup> + (y – b)<sup>2</sup> = r<sup>2</sup>
<br><br>Distance of perpendiculur from (0, 4) to y = x line = r
<br><br>$$ \Rightarrow $$ $$\left| {{{ - b} \over {\sq... | mcq | jee-main-2017-offline | 5,321 |
8nYxtTBHvZiNiZEYJGMbO | maths | circle | tangent-and-normal | The tangent to the circle C<sub>1</sub> : x<sup>2</sup> + y<sup>2</sup> $$-$$ 2x $$-$$ 1 = 0 at the point (2, 1) cuts off a chord of length 4 from a circle C<sub>2</sub> whose center is (3, $$-$$2). The radius of C<sub>2</sub> is : | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "$$\\sqrt 2 $$"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "$$\\sqrt 6 $$"}] | ["D"] | null | Here, equation of tangent on C<sub>1</sub> at (2, 1) is :
<br><br>2x + y $$-$$ (x + 2) $$-$$1 = 0
<br><br>Or x + y = 3
<br><br>If it cuts off the chord of the circle C<sub>2</sub> then the equation of the chord is :
<br>x + y = 3
<br><br>$$\therefore\,\,\,$$ distance of the chord from (3, $$-$$ 2) i... | mcq | jee-main-2018-online-15th-april-evening-slot | 5,323 |
xyLfjlL9Szi7FPwhZZtsV | maths | circle | tangent-and-normal | The tangent and the normal lines at the point
( $$\sqrt 3 $$, 1) to the circle x<sup>2</sup>
+ y<sup>2</sup> = 4 and the x-axis form a triangle. The area of this triangle (in
square units) is : | [{"identifier": "A", "content": "$${4 \\over {\\sqrt 3 }}$$"}, {"identifier": "B", "content": "$${1 \\over {\\sqrt 3 }}$$"}, {"identifier": "C", "content": "$${2 \\over {\\sqrt 3 }}$$"}, {"identifier": "D", "content": "$${1 \\over {3 }}$$"}] | ["C"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266326/exam_images/zc1u5khmyzw8xvgzk5sb.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266898/exam_images/xsjbhlhn0ihvo9jj6lv3.webp"><source media="(max-wid... | mcq | jee-main-2019-online-8th-april-evening-slot | 5,324 |
QDZPAFip0LP89oQqFL18hoxe66ijvwpmdcd | maths | circle | tangent-and-normal | If a tangent to the circle x<sup>2 </sup>+ y<sup>2 </sup> = 1 intersects
the coordinate axes at distinct points P and Q,
then the locus of the mid-point of PQ is : | [{"identifier": "A", "content": "x<sup>2</sup> + y<sup>2</sup> \u2013 4x<sup>2</sup>y<sup>2</sup> = 0"}, {"identifier": "B", "content": "x<sup>2</sup> + y<sup>2</sup> - 2xy = 0"}, {"identifier": "C", "content": "x<sup>2</sup> + y<sup>2</sup> \u2013 2x<sup>2</sup>y<sup>2</sup> = 0"}, {"identifier": "D", "content": "x<su... | ["A"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264900/exam_images/vcvrivi2315vojuhifut.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263280/exam_images/bn8cizbo0h6fmxgzbmhr.webp"><source media="(max-wid... | mcq | jee-main-2019-online-9th-april-morning-slot | 5,325 |
nEOUZMGLBdTQbD83ak3rsa0w2w9jwxveluz | maths | circle | tangent-and-normal | The line x = y touches a circle at the point (1,1). If the circle also passes through the point (1, – 3), then its
radius is : | [{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "2$$\\sqrt 2 $$"}, {"identifier": "D", "content": "3$$\\sqrt 2 $$"}] | ["C"] | null | Equation of circle = (x – 1)<sup>2</sup> + (y –1)<sup>2</sup> + $$\lambda $$(y – x) = 0<br><br>
Which passes through (1, –3)<br><br>
<picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266513/exam_images/n8p3thkkkbmcsid2bddg.webp"><source media="(max-width: 500px)"... | mcq | jee-main-2019-online-10th-april-morning-slot | 5,326 |
gpNR3C5rVO2OLhmiQq7k9k2k5fnh01t | maths | circle | tangent-and-normal | Let the tangents drawn from the origin to the circle, <br/>x<sup>2</sup>
+ y<sup>2</sup>
- 8x - 4y + 16 = 0 touch it at the
points A and B. The (AB)<sup>2</sup>
is equal to : | [{"identifier": "A", "content": "$${{56} \\over 5}$$"}, {"identifier": "B", "content": "$${{32} \\over 5}$$"}, {"identifier": "C", "content": "$${{52} \\over 5}$$"}, {"identifier": "D", "content": "$${{64} \\over 5}$$"}] | ["D"] | null | Equation of chord of contact is
<br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264374/exam_images/gls1xhkqkxbk85dqfbxj.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 7th January Evening Slot Mathematics - Circle Question 101 Eng... | mcq | jee-main-2020-online-7th-january-evening-slot | 5,327 |
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