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JEE Advanced

Circle

Mathematics

Previous Years Questions

MCQ (Single Correct Answer)

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Consider M with $$r = {{({2^{199}} - 1)\sqrt 2 } \over {{2^{198}}}}$$. The number of all those circles Dn that are insid...
JEE Advanced 2021 Paper 2 Online
Consider M with $$r = {{1025} \over {513}}$$. Let k be the number of all those circles Cn that are inside M. Let l be th...
JEE Advanced 2021 Paper 2 Online
Consider a triangle $$\Delta$$ whose two sides lie on the x-axis and the line x + y + 1 = 0. If the orthocenter of $$\De...
JEE Advanced 2021 Paper 1 Online
A line y = mx + 1 intersects the circle $${(x - 3)^2} + {(y + 2)^2}$$ = 25 at the points P and Q. If the midpoint of the...
JEE Advanced 2019 Paper 1 Offline
A tangent PT is drawn to the circle $${x^2}\, + {y^2} = 4$$ at the point P $$\left( {\sqrt 3 ,1} \right)$$. A straight l...
IIT-JEE 2012 Paper 2 Offline
A tangent PT is drawn to the circle $${x^2}\, + {y^2} = 4$$ at the point P $$\left( {\sqrt 3 ,1} \right)$$. A straight l...
IIT-JEE 2012 Paper 2 Offline
The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x - 5y = 20...
IIT-JEE 2012 Paper 1 Offline
The circle passing through the point (-1, 0) and touching the y-axis at (0, 2) also passes through the point.
IIT-JEE 2011 Paper 2 Offline
Tangents drawn from the point P (1, 8) to the circle $${x^2}\, + \,{y^2}\, - \,6x\, - 4y\, - 11 = 0$$ touch the circl...
IIT-JEE 2009
Consider $$\,{L_1}:\,\,2x\,\, + \,\,3y\, + \,p\,\, - \,\,3 = 0$$ $$\,{L_2}:\,\,2x\,\, + \,\,3y\, + \,p\,\, + \,\,3 = 0...
IIT-JEE 2008
A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP...
IIT-JEE 2008
A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP...
IIT-JEE 2008
A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP...
IIT-JEE 2008
ABCD is a square of side length 2 units. $${C_1}$$ is the circle touching all the sides of the square ABCD and $${C_2}$$...
IIT-JEE 2006
ABCD is a square of side length 2 units. $$C_1$$ is the circle touching all the sides of the square ABCD and $$C_2$$ is ...
IIT-JEE 2006
ABCD is a square of side length 2 units. $$C_1$$ is the circle touching all the sides of the square ABCD and $$C_2$$ is ...
IIT-JEE 2006
A circle is given by $${x^2}\, + \,{(y\, - \,1\,)^2}\, = \,1$$, another circle C touches it externally and also the x-a...
IIT-JEE 2005 Screening
If one of the diameters of the circle $${x^2} + {y^2} - 2x - 6y + 6 = 0$$ is a chord to the circle with centre (2, 1), t...
IIT-JEE 2004 Screening
The centre of circle inscibed in square formed by the lines $${x^2} - 8x + 12 = 0\,\,and\,{y^2} - 14y + 45 = 0$$, is
IIT-JEE 2003 Screening
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 ...
IIT-JEE 2002 Screening
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 poi...
IIT-JEE 2001 Screening
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...
IIT-JEE 2001 Screening
If the circles $${x^2}\, + \,{y^2}\, + \,\,2x\, + \,2\,k\,y\,\, + \,6\,\, = \,\,0,\,\,{x^2}\, + \,\,{y^2}\, + \,2ky\, + ...
IIT-JEE 2000 Screening
The triangle PQR is inscribed in the circle $${x^2}\, + \,\,{y^2} = \,25$$. If Q and R have co-ordinates (3, 4) and ( -...
IIT-JEE 2000 Screening
If two distinct chords, drawn from the point (p, q) on the circle $${x^2}\, + \,{y^2} = \,px\, + \,qy\,\,(\,where\,pq\, ...
IIT-JEE 1999
The angle between a pair of tangents drawn from a point P to the circle $${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\, + \,...
IIT-JEE 1996
The circles $${x^2} + {y^2} - 10x + 16 = 0$$ and $${x^2} + {y^2} = {r^2}$$ intersect each other in two distinct points i...
IIT-JEE 1994
The locus of the centre of a circle, which touches externally the circle $${x^2} + {y^2} - 6x - 6y + 14 = 0$$ and also t...
IIT-JEE 1993
The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle $${x^2} + {y^2} = 9$$is
IIT-JEE 1992
The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle of area 154 sq. units. Then the equation of this circle ...
IIT-JEE 1989
If the two circles $${(x - 1)^2} + {(y - 3)^2} = {r^2}$$ and $${x^2} + {y^2} - 8x + 2y + 8 = 0$$ intersect in two distin...
IIT-JEE 1989
If a circle passes through the point (a, b) and cuts the circle $${x^2}\, + \,{y^2}\, = \,{k^2}$$ orthogonally, then the...
IIT-JEE 1988
The locus of the mid-point of a chord of the circle $${x^2} + {y^2} = 4$$ which subtends a right angle at the origin is
IIT-JEE 1984
The equation of the circle passing through (1, 1) and the points of intersection of $${x^2} + {y^2} + 13x - 3y = 0$$ an...
IIT-JEE 1983
The centre of the circle passing through the point (0, 1) and touching the curve $$\,y = {x^2}$$ at (2, 4) is
IIT-JEE 1983
Two circles $${x^2} + {y^2} = 6$$ and $${x^2} + {y^2} - 6x + 8 = 0$$ are given. Then the equation of the circle through ...
IIT-JEE 1980
A square is inscribed in the circle $${x^2} + {y^2} - 2x + 4y + 3 = 0$$. Its sides are parallel to the coordinate axes. ...
IIT-JEE 1980

Numerical

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Consider the region R = {(x, y) $$\in$$ R $$\times$$ R : x $$\ge$$ 0 and y2 $$\le$$ 4 $$-$$ x}. Let F be the family of a...
JEE Advanced 2021 Paper 2 Online
Consider the region R = {(x, y) $$\in$$ R $$\times$$ R : x $$\ge$$ 0 and y2 $$\le$$ 4 $$-$$ x}. Let F be the family of a...
JEE Advanced 2021 Paper 2 Online
Let O be the centre of the circle x2 + y2 = r2, where $$r > {{\sqrt 5 } \over 2}$$. Suppose PQ is a chord of this cir...
JEE Advanced 2020 Paper 2 Offline
Let the point B be the reflection of the point A(2, 3) with respect to the line $$8x - 6y - 23 = 0$$. Let $$\Gamma_{A} $...
JEE Advanced 2019 Paper 1 Offline
The straight line 2x - 3y = 1 divides the circular region $${x^2}\, + \,{y^2}\, \le \,6$$ into two parts. If $$S = \lef...
IIT-JEE 2011 Paper 2 Offline
The centres of two circles $${C_1}$$ and $${C_2}$$ each of unit radius are at a distance of 6 units from each other. Let...
IIT-JEE 2009

MCQ (More than One Correct Answer)

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Let RS be the diameter of the circle $${x^2}\, + \,{y^2} = 1$$, where S is the point (1, 0). Let P be a variable point (...
JEE Advanced 2016 Paper 1 Offline
A circle S passes through the point (0, 1) and is orthogonal to the circles $${(x - 1)^2}\, + \,{y^2} = 16\,\,and\,\,{x^...
JEE Advanced 2014 Paper 1 Offline
Circle (s) touching x-axis at a distance 3 from the origin and having an intercept of length $$2\sqrt 7 $$ on y-axis is ...
JEE Advanced 2013 Paper 2 Offline
The number of common tangents to the circles $${x^2}\, + \,{y^2} = 4$$ and $${x^2}\, + \,{y^2}\, - 6x\, - 8y = 24$$ is
IIT-JEE 1998
If the circle $${x^2}\, + \,{y^2} = \,{a^2}$$ intersects the hyperbola $$xy = {c^2}$$ in four points $$P\,({x_1},\,{y_1}...
IIT-JEE 1998
The equations of the tangents drawn from the origin to the circle $${x^2}\, + \,{y^2}\, - \,2rx\,\, - 2hy\, + {h^2} = 0$...
IIT-JEE 1988

Subjective

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Circles with radii 3, 4 and 5 touch each other externally. It P is the point of intersection of tangents to these circle...
IIT-JEE 2005
Find the equation of circle touching the line 2x + 3y + 1 = 0 at (1, -1) and cutting orthogonally the circle having line...
IIT-JEE 2004
For the circle $${x^2}\, + \,{y^2} = {r^2}$$, find the value of r for which the area enclosed by the tangents drawn from...
IIT-JEE 2003
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 ra...
IIT-JEE 2001
Let $$C_1$$ and $$C_2$$ be two circles with $$C_2$$ lying inside $$C_1$$. A circle C lying inside $$C_1$$ touches $$C_1$...
IIT-JEE 2001
Let $${T_1}$$, $${T_2}$$ be two tangents drawn from (- 2, 0) onto the circle $$C:{x^2}\,\, + \,{y^2} = 1$$. Determine th...
IIT-JEE 1999
$$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...
IIT-JEE 1998
Let C be any circle with centre $$\,\left( {0\, , \sqrt {2} } \right)$$. Prove that at the most two rational points can ...
IIT-JEE 1997
Find the intervals of value of a for which the line y + x = 0 bisects two chords drawn from a point $$\left( {{{1\, + \,...
IIT-JEE 1996
A circle passes through three points A, B and C with the line segment AC as its diameter. A line passing through A angle...
IIT-JEE 1996
Consider a family of circles passing through two fixed points A (3, 7) and B (6, 5). Show that the chords on which the c...
IIT-JEE 1993
Find the coordinates of the point at which the circles $${x^2}\, + \,{y^2} - \,4x - \,2y = - 4\,\,and\,\,{x^2}\, + \,{y...
IIT-JEE 1993
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 ...
IIT-JEE 1992
Two circles, each of radius 5 units, touch each other at (1, 2). If the equation of their common tangent is 4x + 3y = 10...
IIT-JEE 1991
A circle touches the line y = x at a point P such that OP = $${4\sqrt 2 \,}$$, where O is the origin. The circle contain...
IIT-JEE 1990
If $$\left( {{m_i},{1 \over {{m_i}}}} \right),\,{m_i}\, > \,0,\,i\, = 1,\,2,\,3,\,4$$ are four distinct points on a c...
IIT-JEE 1989
Let a given line $$L_1$$ intersects the x and y axes at P and Q, respectively. Let another line $$L_2$$, perpendicular t...
IIT-JEE 1987
The circle $${x^2}\, + \,{y^2} - \,4x\, - 4y + \,4 = 0$$ is inscribed in a triangle which has two of its sides along the...
IIT-JEE 1987
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...
IIT-JEE 1986
The abscissa of the two points A and B are the roots of the equation $${x^2}\, + \,2ax\, - {b^2} = 0$$ and their ordinat...
IIT-JEE 1984
Through a fixed point (h, k) secants are drawn to the circle $$\,{x^2}\, + \,{y^2} = \,{r^2}$$. Show that the locus of t...
IIT-JEE 1983
Find the equations of the circle passing through (- 4, 3) and touching the lines x + y = 2 and x - y = 2.
IIT-JEE 1981
Let A be the centre of the circle $${x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, - 20 = 0\,$$. Suppose that the tangents at the...
IIT-JEE 1981
Find the equation of the circle whose radius is 5 and which touches the circle $${x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, -...
IIT-JEE 1978

Fill in the Blanks

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The chords of contact of the pair of tangents drawn from each point on the line 2x + y = 4 to circle $${x^2} + {y^2} = 1...
IIT-JEE 1997
For each natural number k, let $${C_k}$$ denote the circle with radius k centimetres and centre at the origin. On the ci...
IIT-JEE 1997
The intercept on the line y = x by the circle $${x^2} + {y^2} - 2x = 0$$ is AB. Equation of the circle with AB as a dia...
IIT-JEE 1996
The equation of the locus of the mid-points of the circle $$4{x^2} + 4{y^2} - 12x + 4y + 1 = 0$$ that subtend an angle ...
IIT-JEE 1993
If a circle passes through the points of intersection of the coordinate axes with the lines $$\lambda \,x - y + 1 = 0$$ ...
IIT-JEE 1991
The area of the triangle formed by the positive x-axis and the normal and the tangent to the circle $${x^2} + {y^2} = 4\...
IIT-JEE 1989
If the circle $${C_1}:{x^2} + {y^2} = 16$$ intersects another circle $${C_2}$$ of radius 5 in such a manner that common ...
IIT-JEE 1988
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 ...
IIT-JEE 1987
The equation of the line passing through the points of intersection of the circles $$3{x^2} + 3{y^2} - 2x + 12y - 9 = 0$...
IIT-JEE 1986
From the point A(0, 3) on the circle $${x^2} + 4x + {(y - 3)^2} = 0$$, a chord AB is drawn and extended to a point M suc...
IIT-JEE 1986
Let $${x^2} + {y^2} - 4x - 2y - 11 = 0$$ be a circle. A pair of tangentas from the point (4, 5) with a pair of radi from...
IIT-JEE 1985
From the origin chords are drawn to the circle $${(x - 1)^2} + {y^2} = 1$$. The equation of the locus of the mid-points ...
IIT-JEE 1985
The lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to the same circle. The radius of this circle is ...............
IIT-JEE 1984
The point of intersection of the line 4x - 3y - 10 = 0 and the circle $${x^2} + {y^2} - 2x + 4y - 20 = 0$$ are ............
IIT-JEE 1983
If A and B are points in the plane such that PA/PB = k (constant) for all P on a given circle, then the value of k canno...
IIT-JEE 1982

True or False

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The line x + 3y = 0 is a diameter of the circle $${x^2} + {y^2} - 6x + 2y = 0\,$$.
IIT-JEE 1989
No tangent can be drawn from the point (5/2, 1) to the circumcircle of the triangle with vertices $$\left( {1,\sqrt 3 } ...
IIT-JEE 1985

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