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MCQ (More than One Correct Answer)

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Let $G$ be a circle of radius $R>0$. Let $G_{1}, G_{2}, \ldots, G_{n}$ be $n$ circles of equal radius $r>0$. Suppose eac...
JEE Advanced 2022 Paper 2 Online
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 $$on y-axis is ... JEE Advanced 2013 Paper 2 Offline If the circle$${x^2}\, + \,{y^2} = \,{a^2}$$intersects the hyperbola$$xy = {c^2}$$in four points$$P\,({x_1},\,{y_1}...
IIT-JEE 1998
The number of common tangents to the circles $${x^2}\, + \,{y^2} = 4$$ and $${x^2}\, + \,{y^2}\, - 6x\, - 8y = 24$$ is
IIT-JEE 1998
The equations of the tangents drawn from the origin to the circle $${x^2}\, + \,{y^2}\, - \,2rx\,\, - 2hy\, + {h^2} = 0... IIT-JEE 1988 Numerical More Let$$A B C$$be the triangle with$$A B=1, A C=3$$and$$\angle B A C=\frac{\pi}{2}$$. If a circle of radius$$r>0$$to... JEE Advanced 2022 Paper 1 Online Consider the region R = {(x, y)$$\in$$R$$\times$$R : x$$\ge$$0 and y2$$\le$$4$$-$$x}. Let F be the family of a... JEE Advanced 2021 Paper 2 Online Consider the region R = {(x, y)$$\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 &gt; {{\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}$$each of unit radius are at a distance of 6 units from each other. Let... IIT-JEE 2009 MCQ (Single Correct Answer) More Consider M with$$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 Subjective More 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 $$\left( {{m_i},{1 \over {{m_i}}}} \right),\,{m_i}\, &gt; \,0,\,i\, = 1,\,2,\,3,\,4$$ are four distinct points on a c... IIT-JEE 1989 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 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 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 $${x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, -... IIT-JEE 1978 Fill in the Blanks More The chords of contact of the pair of tangents drawn from each point on the line 2x + y = 4 to circle$${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
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
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
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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