Circle
615 Questions
Start JEE Mains Test
2003
Q551
JEE Advanced
Numerical
14 Mar 2026
For the circle ${x^2}\, + \,{y^2} = {r^2}$, find the value of r for which the area enclosed by the tangents drawn from the point P (6, 8) to the circle and the chord of contact is maximum.
Correct Answer: 5
2002
Q552
JEE Mains
MCQ
14 Mar 2026
The centres of a set of circles, each of radius 3, lie on the circle ${x^2}\, + \,{y^2} = 25$. The locus of any point in the set is :
A.
$4\, \le \,\,{x^2}\, + \,{y^2}\, \le \,\,64$
B.
${x^2}\, + \,{y^2}\, \le \,\,25$
C.
${x^2}\, + \,{y^2}\, \ge \,\,25$
D.
$3\, \le \,\,{x^2}\, + \,{y^2}\, \le \,\,9$
2002
Q553
JEE Mains
MCQ
14 Mar 2026
If the chord y = mx + 1 of the circle ${x^2}\, + \,{y^2} = 1$ subtends an angle of measure ${45^ \circ }$ at the major segment of the circle then value of m is :
A.
$2\, \pm \,\sqrt 2 \,\,$
B.
$ - \,2\, \pm \,\sqrt 2 \,$
C.
$- 1\, \pm \,\sqrt 2 \,\,$
D.
none of these
2002
Q554
JEE Mains
MCQ
14 Mar 2026
The equation of a circle with origin as a center and passing through an equilateral triangle whose median is of length $3$$a$ is :
A.
${x^2}\, + \,{y^2} = 9{a^2}$
B.
${x^2}\, + \,{y^2} = 16{a^2}$
C.
${x^2}\, + \,{y^2} = 4{a^2}$
D.
${x^2}\, + \,{y^2} = {a^2}$
2002
Q555
JEE Mains
MCQ
14 Mar 2026
The centre of the circle passing through (0, 0) and (1, 0) and touching the circle ${x^2}\, + \,{y^2} = 9$ is :
A.
$\left( {{1 \over 2},\,{1 \over 2}} \right)$
B.
$\left( {{1 \over 2},\, - \,\sqrt 2 } \right)$
C.
$\left( {{3 \over 2},\,{1 \over 2}} \right)$
D.
$\left( {{1 \over 2},\,{3 \over 2}} \right)$
2002
Q556
JEE Advanced
MCQ
14 Mar 2026
If the tangent at the point P on the circle ${x^2} + {y^2} + 6x + 6y = 2$ meets a straight line 5x - 2y + 6 = 0 at a point Q on the y-axis, then the lenght of PQ is
A.
4
B.
${2\sqrt 5 }$
C.
5
D.
${3\sqrt 5 }$
2002
Q557
JEE Advanced
MCQ
14 Mar 2026
If $a > 2b > 0$ then the positive value of $m$ for which $y = mx - b\sqrt {1 + {m^2}} $ is a common tangent to ${x^2} + {y^2} = {b^2}$ and ${\left( {x - a} \right)^2} + {y^2} = {b^2}$ is
A.
${{2b} \over {\sqrt {{a^2} - 4{b^2}} }}$
B.
${{\sqrt {{a^2} - 4{b^2}} } \over {2b}}$
C.
${{2b} \over {a - 2b}}$
D.
${{b} \over {a - 2b}}$
2001
Q558
JEE Advanced
MCQ
14 Mar 2026
Let A B be a chord of the circle ${x^2} + {y^2} = {r^2}$ subtending a right angle at the centre. Then the locus of the centriod of the triangle PAB as P moves on the circle is
A.
a parabola
B.
a circle
C.
an ellipse
D.
a pair of straight lines
2001
Q559
JEE Advanced
MCQ
14 Mar 2026
Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals
A.
$\sqrt {PQ.\,RS} $
B.
(PQ + RS) / 2
C.
2 PQ. RS/(PQ + RS)
D.
$\sqrt {\left( {P{Q^2} + \,R{S^2}} \right)} \,\,/2$
2001
Q560
JEE Advanced
Numerical
14 Mar 2026
Let $C_1$ and $C_2$ be two circles with $C_2$ lying inside $C_1$. A circle C lying inside $C_1$ touches $C_1$ internally and $C_2$ externally. Identify the locus of the centre of C.
Correct Answer: ellipse
2001
Q561
JEE Advanced
Numerical
14 Mar 2026
Let $\,2{x^2}\, + \,{y^2} - \,3xy = 0$ be the equation of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA.
Correct Answer: $$3\,\left( {3\, + \sqrt {10} } \right)$$
2000
Q562
JEE Advanced
MCQ
14 Mar 2026
If the circles ${x^2}\, + \,{y^2}\, + \,\,2x\, + \,2\,k\,y\,\, + \,6\,\, = \,\,0,\,\,{x^2}\, + \,\,{y^2}\, + \,2ky\, + \,k\, = \,0$ intersect orthogonally, then k is
A.
2 or $ - {3 \over 2}$
B.
- 2 or $ - {3 \over 2}$
C.
2 or $ {3 \over 2}$
D.
- 2 or $ {3 \over 2}$
2000
Q563
JEE Advanced
MCQ
14 Mar 2026
The triangle PQR is inscribed in the circle ${x^2}\, + \,\,{y^2} = \,25$. If Q and R have co-ordinates (3, 4) and ( - 4, 3) respectively, then $\angle \,Q\,P\,R$ is equal to
A.
${\pi \over 2}$
B.
${\pi \over 3}$
C.
${\pi \over 4}$
D.
${\pi \over 6}$
1999
Q564
JEE Advanced
MCQ
14 Mar 2026
If two distinct chords, drawn from the point (p, q) on the circle ${x^2}\, + \,{y^2} = \,px\, + \,qy\,\,(\,where\,pq\, \ne \,0)$ are bisected by the x - axis, then
A.
${p^2}\, = \,\,{q^2}$
B.
$\,{p^2}\, = \,\,8\,{q^2}$
C.
${p^2}\, < \,\,8\,{q^2}$
D.
${p^2}\, > \,\,8\,{q^2}$.
1999
Q565
JEE Advanced
Numerical
14 Mar 2026
Let ${T_1}$, ${T_2}$ be two tangents drawn from (- 2, 0) onto the circle $C:{x^2}\,\, + \,{y^2} = 1$. Determine the circles touching C and having ${T_1}$, ${T_2}$ as their pair of tangents. Further, find the equations of all possible common tangents to these circles, when taken two at a time.
Correct Answer: <br> $${(x - y)^2} + \,{y^2} = {3^2}\,\,\,and\,\,{\left( {x + \,{4 \over 3}} \right)^2} + \,{y^2} = \,{\left( {{1 \over 3}} \right)^2};$$
<br> $$y = \pm \,{5 \over {\sqrt {39} \,}}\left( {x + {4 \over 5}} \right)$$
1998
Q566
JEE Advanced
MCQ
14 Mar 2026
The number of common tangents to the circles ${x^2}\, + \,{y^2} = 4$ and ${x^2}\, + \,{y^2}\, - 6x\, - 8y = 24$ is
A.
0
B.
1
C.
3
D.
4
1998
Q567
JEE Advanced
MSQ
14 Mar 2026
If the circle ${x^2}\, + \,{y^2} = \,{a^2}$ intersects the hyperbola $xy = {c^2}$ in four points $P\,({x_1},\,{y_1}),\,Q\,\,({x_2},\,{y_2}),\,\,R\,({x_3},\,{y_3}),\,S\,({x_4},\,{y_4}),$ then
A.
${x_1}\, + \,{x_2} + \,{x_3}\, + \,{x_4}\, = 0$
B.
${y_1}\, + \,{y_2} + \,{y_3}\, + \,{y_4}\, = 0$
C.
${x_1}\,{x_2}\,{x_3}\,{x_4}\, = {c^4}$
D.
${y_1}\,{y_2}\,{y_3}\,{y_4}\, = {c^4}$
1998
Q568
JEE Advanced
Numerical
14 Mar 2026
$C_1$ and $C_2$ are two concentric circles, the radius of $C_2$ being twice that of $C_1$. From a point P on $C_2$, tangents PA and PB are drawn to $C_1$. Prove that the centroid of the triangle PAB lies on $C_1$.
Correct Answer: solve it
1997
Q569
JEE Advanced
Numerical
14 Mar 2026
Let C be any circle with centre $\,\left( {0\, , \sqrt {2} } \right)$. Prove that at the most two rational points can to there on C. (A rational point is a point both of whose coordinates are rational numbers.)
Correct Answer: solve it
1997
Q570
JEE Advanced
Numerical
14 Mar 2026
The chords of contact of the pair of tangents drawn from each point on the line 2x + y = 4 to circle ${x^2} + {y^2} = 1$ pass through the point........................
Correct Answer: $$\left( {{1 \over 2},{1 \over 4}} \right)$$
1997
Q571
JEE Advanced
Numerical
14 Mar 2026
For each natural number k, let ${C_k}$ denote the circle with radius k centimetres and centre at the origin. On the circle ${C_k}$, a-particle moves k centimetres in the counter-clockwise direction. After completing its motion on ${C_k}$, the particle moves to ${C_{k + 1}}$ in the radial direction. The motion of the patticle continues in the manner. The particle starts at (1, 0). If the particle crosses the positive direction of the x-axis for the first time on the circle ${C_n}$ then n = ..............
Correct Answer: 7
1996
Q572
JEE Advanced
MCQ
14 Mar 2026
The angle between a pair of tangents drawn from a point P to the circle ${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\, + \,9\,{\sin ^2}\,\alpha \, + \,13\,{\cos ^2}\,\alpha \, = \,0$ is $2\,\alpha $.
The equation of the locus of the point P is
The equation of the locus of the point P is
A.
${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\, + \,4\, = \,0$
B.
${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\,\, - \,9\,\, = \,0$
C.
${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\,\, - \,4\,\, = \,0$
D.
${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\,\, + \,9\,\, = \,0$
1996
Q573
JEE Advanced
Numerical
14 Mar 2026
Find the intervals of value of a for which the line y + x = 0 bisects two chords drawn from a point $\left( {{{1\, + \,\sqrt 2 a} \over 2},\,{{1\, - \,\sqrt 2 a} \over 2}} \right)$ to the circle $\,\,2{x^2}\, + \,2{y^2} - (\,1\, + \sqrt 2 a)\,x - (1 - \sqrt 2 a)\,y = 0$.
Correct Answer: $$a\, \in \,]\, - \infty ,\,\, - \,2\,[U]\,\,2,\,\,\infty \,[$$
1996
Q574
JEE Advanced
Numerical
14 Mar 2026
A circle passes through three points A, B and C with the line segment AC as its diameter. A line passing through A angles DAB and CAB are $\,\alpha \,\,and\,\,\beta $ respectively and the distance between the point A and the mid point of the line segment DC is d, prove that the area of the circle is $${{\pi \,{d^2}\,\,{{\cos }^2}\,\,\alpha } \over {{{\cos }^2}\,\alpha \, + \,{{\cos }^2}\,\beta \, + \,\,2\,\cos \,\,\alpha \,\,\cos \,\beta \,\cos \,\,(\beta - \alpha )\,}}$$
Correct Answer: solve it
1996
Q575
JEE Advanced
Numerical
14 Mar 2026
The intercept on the line y = x by the circle ${x^2} + {y^2} - 2x = 0$ is AB. Equation of the circle with AB as a diameter is................................
Correct Answer: $${x^2} + {y^2} - x - y = 0$$
1994
Q576
JEE Advanced
MCQ
14 Mar 2026
The circles ${x^2} + {y^2} - 10x + 16 = 0$ and ${x^2} + {y^2} = {r^2}$ intersect each other in two distinct points if
A.
r < 2
B.
r > 8
C.
2 < r < 8
D.
$2 \le r \le 8$
1993
Q577
JEE Advanced
MCQ
14 Mar 2026
The locus of the centre of a circle, which touches externally the circle ${x^2} + {y^2} - 6x - 6y + 14 = 0$ and also touches the y-axis, is given by the equation:
A.
${x^2} - 6x - 10y + 14 = 0$
B.
${x^2} - 10x - 6y + 14 = 0$
C.
${y^2} - 6x - 10y + 14 = 0$
D.
${y^2} - 10x - 6y + 14 = 0$
1993
Q578
JEE Advanced
Numerical
14 Mar 2026
Find the coordinates of the point at which the circles ${x^2}\, + \,{y^2} - \,4x - \,2y = - 4\,\,and\,\,{x^2}\, + \,{y^2} - \,12x - \,8y = - 36$ touch each other. Also find equations common tangests touching the circles in the distinct points.
Correct Answer: $$\left( {{{14} \over 5},{8 \over 5}} \right),\,\,y = 0$$ and $$7y - \,24x\, + \,16\, = 0$$
1993
Q579
JEE Advanced
Numerical
14 Mar 2026
Consider a family of circles passing through two fixed points A (3, 7) and B (6, 5). Show that the chords on which the circle ${x^2}\, + \,{y^2} - \,4x - \,6y - 3 = 0$ cuts the members of the family are concurrent at a point. Find the coordinate of this point.
Correct Answer: $$\left( {2\,,{{23} \over 3}} \right)$$
1993
Q580
JEE Advanced
Numerical
14 Mar 2026
The equation of the locus of the mid-points of the circle $4{x^2} + 4{y^2} - 12x + 4y + 1 = 0$ that subtend an angle of $2\pi /3$ at its centre is.................................
Correct Answer: $$16{x^2} + 16{y^2} - 48x + 16y + 31 = 0$$
1992
Q581
JEE Advanced
MCQ
14 Mar 2026
The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle ${x^2} + {y^2} = 9$is
A.
$\left( {{3 \over 2},{1 \over 2}} \right)\,$
B.
$\left( {{1 \over 2},{3 \over 2}} \right)\,$
C.
$\left( {{1 \over 2},{1 \over 2}} \right)\,$
D.
$\left( {{1 \over 2}, - {2^{{1 \over 2}}}} \right)\,$
1992
Q582
JEE Advanced
Numerical
14 Mar 2026
Let a circle be given by 2x (x - a) + y (2y - b) = 0, $(a\, \ne \,0,\,\,b\, \ne 0)$. Find the condition on a abd b if two chords, each bisected by the x-axis, can be drawn to the circle from $\left( {a,\,\,{b \over 2}} \right)$.
Correct Answer: $${a^2}\, > \,2\,{b^2}$$
1991
Q583
JEE Advanced
Numerical
14 Mar 2026
Two circles, each of radius 5 units, touch each other at (1, 2). If the equation of their common tangent is 4x + 3y = 10, find the equation of the circles.
Correct Answer: <br> $${x^2}\, + \,{y^2} + 6x\, + 2y - 15\, = 0$$ and
<br> $${x^2}\, + \,{y^2} - 10x\, - 10y + 25\, = 0$$
1991
Q584
JEE Advanced
Numerical
14 Mar 2026
If a circle passes through the points of intersection of the coordinate axes with the lines $\lambda \,x - y + 1 = 0$ and x - 2y + 3 = 0, then the value of $\lambda $ = .........
Correct Answer: 2
1990
Q585
JEE Advanced
Numerical
14 Mar 2026
A circle touches the line y = x at a point P such that OP = ${4\sqrt 2 \,}$, where O is the origin. The circle contains the point (- 10, 2) in its interior and the length of its chord on the line x + y = 0 is ${6\sqrt 2 \,}$. Determine the equation of the circle.
Correct Answer: $${x^2}\, + {y^2} + \,18x\, - 2y\, + 32\, = 0$$
1989
Q586
JEE Advanced
MCQ
14 Mar 2026
The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle of area 154 sq. units. Then the equation of this circle is
A.
${x^2} + {y^2} + 2x - 2y = 62$
B.
${x^2} + {y^2} + 2x - 2y = 47$
C.
${x^2} + {y^2} - 2x + 2y = 47$
D.
${x^2} + {y^2} - 2x + 2y = 62$c
1989
Q587
JEE Advanced
MCQ
14 Mar 2026
If the two circles ${(x - 1)^2} + {(y - 3)^2} = {r^2}$ and ${x^2} + {y^2} - 8x + 2y + 8 = 0$ intersect in two distinct points, then
A.
2 < r < 8
B.
r < 2
C.
r = 2
D.
r > 2
1989
Q588
JEE Advanced
Numerical
14 Mar 2026
If $\left( {{m_i},{1 \over {{m_i}}}} \right),\,{m_i}\, > \,0,\,i\, = 1,\,2,\,3,\,4$ are four distinct points on a circle, then show that ${m_1}\,{m_2}\,{m_3}\,{m_4}\, = 1$
Correct Answer: solve it
1989
Q589
JEE Advanced
Numerical
14 Mar 2026
The area of the triangle formed by the positive x-axis and the normal and the tangent to the circle ${x^2} + {y^2} = 4\,\,at\,\,\left( {1,\sqrt 3 } \right)$ is,..................
Correct Answer: $${2\sqrt 3 }$$ sq unit
1989
Q590
JEE Advanced
MCQ
14 Mar 2026
The line x + 3y = 0 is a diameter of the circle ${x^2} + {y^2} - 6x + 2y = 0\,$.
A.
TRUE
B.
FALSE
1988
Q591
JEE Advanced
MCQ
14 Mar 2026
If a circle passes through the point (a, b) and cuts the circle ${x^2}\, + \,{y^2}\, = \,{k^2}$ orthogonally, then the equation of the locus of its centre is
A.
$2\,ax\, + \,2\,by\, - \,({a^2}\, + \,{b^2}\, + \,\,{k^2})\, = \,0$
B.
$2\,ax\, + \,2\,by\, - \,({a^2}\, - \,\,{b^2}\, + \,\,{k^2})\, = \,0$
C.
${x^2}\, + \,{y^2}\, - \,3\,\,ax\, + \,4\,by\, + \,\,({a^2}\, + \,\,{b^2}\, - \,\,{k^2})\, = \,0$
D.
${x^2}\, + \,{y^2}\, - \,2\,\,ax\, - \,4\,by\, + \,\,({a^2}\, - \,\,{b^2}\, - \,\,{k^2})\, = \,0$.
1988
Q592
JEE Advanced
MSQ
14 Mar 2026
The equations of the tangents drawn from the origin to the circle ${x^2}\, + \,{y^2}\, - \,2rx\,\, - 2hy\, + {h^2} = 0$, are
A.
x = 0
B.
y = 0
C.
$({h^2}\, - \,{r^2})\,x - \,\,2rhy\, = \,0$
D.
$({h^2}\, - \,{r^2})\,x + \,\,2rhy\, = \,0$
1988
Q593
JEE Advanced
Numerical
14 Mar 2026
If the circle ${C_1}:{x^2} + {y^2} = 16$ intersects another circle ${C_2}$ of radius 5 in such a manner that common chord is of maximum lenght and has a slope equal to 3/4, then the coordinates of the centre of ${C_2}$ are.............................
Correct Answer: $$\,\left( { - {9 \over 5},{{12} \over 5}} \right)\,\,or\,\left( {{9 \over 5},\, - {{12} \over 5}} \right)$$
1987
Q594
JEE Advanced
Numerical
14 Mar 2026
Let a given line $L_1$ intersects the x and y axes at P and Q, respectively. Let another line $L_2$, perpendicular to $L_1$, cut the x and y axes at R and S, respectively. Show that the locus of the point of intersection of the lines PS and QR is a circle passing through the origin.
Correct Answer: solve it
1987
Q595
JEE Advanced
Numerical
14 Mar 2026
The circle ${x^2}\, + \,{y^2} - \,4x\, - 4y + \,4 = 0$ is inscribed in a triangle which has two of its sides along the co-ordinate axes. The locus of the circumcentre of the triangle is $x\, + \,y\, - xy\, + k\,{\left( {{x^2}\, + \,{y^2}} \right)^{1/2}} = 0$. Find k.
Correct Answer: k = 1
1987
Q596
JEE Advanced
Numerical
14 Mar 2026
The area of the triangle formed by the tangents from the point (4, 3) to the circle ${x^2} + {y^2} = 9$ and the line joining their points of contact is...................
Correct Answer: $${{192} \over {25}}$$
1986
Q597
JEE Advanced
Numerical
14 Mar 2026
Lines 5x + 12y - 10 = 0 and 5x - 12y - 40 = 0 touch a circle $C_1$ of diameter 6. If the centre of $C_1$ lies in the first quadrant, find the equation of the circle $C_2$ which is concentric with $C_1$ and cuts intercepts of length 8 on these lines.
Correct Answer: $${x^2}\, + \,{y^2} - \,10x\, - 4y + \,4 = 0\,$$
1986
Q598
JEE Advanced
Numerical
14 Mar 2026
The equation of the line passing through the points of intersection of the circles $3{x^2} + 3{y^2} - 2x + 12y - 9 = 0$ and ${x^2} + {y^2} - 6x + 2y - 15 = 0$ is..............................
Correct Answer: 10x - 3y - 18 = 0
1986
Q599
JEE Advanced
Numerical
14 Mar 2026
From the point A(0, 3) on the circle ${x^2} + 4x + {(y - 3)^2} = 0$, a chord AB is drawn and extended to a point M such that AM = 2AB. The equation of the locus of M is..........................
Correct Answer: $${x^2} + {y^2} + 8x - 6y + 9 = 0$$
1985
Q600
JEE Advanced
Numerical
14 Mar 2026
From the origin chords are drawn to the circle ${(x - 1)^2} + {y^2} = 1$. The equation of the locus of the mid-points of these chords is.............
Correct Answer: $${x^2} + {y^2} - x = 0$$