Mathematics
Circle
Previous Years Questions

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 each of the $n$ circles $G_{1}, G... 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 (other than R and S) on the cir... A circle S passes through the point (0, 1) and is orthogonal to the circles $${(x - 1)^2}\, + \,{y^2} = 16\,\,and\,\,{x^2}\, + \,{y^2} = 1$$. Then 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 (are) If the circle $${x^2}\, + \,{y^2} = \,{a^2}$$ intersects the hyperbola $$xy = {c^2}$$ in four points $$P\,({x_1},\,{y_1}),\,Q\,\,({x_2},\,{y_2}),\,\,R... The number of common tangents to the circles$${x^2}\, + \,{y^2} = 4$$and$${x^2}\, + \,{y^2}\, - 6x\, - 8y = 24$$is The equations of the tangents drawn from the origin to the circle$${x^2}\, + \,{y^2}\, - \,2rx\,\, - 2hy\, + {h^2} = 0$$, are ## Numerical Let$$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$$touches the sides$$A B, A C$$a... Consider the region R = {(x, y)$$\in$$R$$\times$$R : x$$\ge$$0 and y2$$\le$$4$$-$$x}. Let F be the family of all circles that are contained ... Consider the region R = {(x, y)$$\in$$R$$\times$$R : x$$\ge$$0 and y2$$\le$$4$$-$$x}. Let F be the family of all circles that are contained ... Let O be the centre of the circle x2 + y2 = r2, where$$r > {{\sqrt 5 } \over 2}$$. Suppose PQ is a chord of this circle and the equation of the li... Let the point B be the reflection of the point A(2, 3) with respect to the line$$8x - 6y - 23 = 0$$. Let$$\Gamma_{A} $$and$$\Gamma_{B} $$be circl... The straight line 2x - 3y = 1 divides the circular region$${x^2}\, + \,{y^2}\, \le \,6$$into two parts. If$$S = \left\{ {\left( {2,\,{3 \over 4}} ... 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 P be the mid point of the lin... ## MCQ (Single Correct Answer) Consider M with $$r = {{1025} \over {513}}$$. Let k be the number of all those circles Cn that are inside M. Let l be the maximum possible number of c... Consider M with $$r = {{({2^{199}} - 1)\sqrt 2 } \over {{2^{198}}}}$$. The number of all those circles Dn that are inside M is Consider a triangle $$\Delta$$ whose two sides lie on the x-axis and the line x + y + 1 = 0. If the orthocenter of $$\Delta$$ is (1, 1), then the equa... 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 line segment PQ has x-coordin... A tangent PT is drawn to the circle $${x^2}\, + {y^2} = 4$$ at the point P $$\left( {\sqrt 3 ,1} \right)$$. A straight line L, perpendicular to PT is ... A tangent PT is drawn to the circle $${x^2}\, + {y^2} = 4$$ at the point P $$\left( {\sqrt 3 ,1} \right)$$. A straight line L, perpendicular to PT is ... The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x - 5y = 20 to the circle $${x^2}\, + \,{... The circle passing through the point (-1, 0) and touching the y-axis at (0, 2) also passes through the point. Tangents drawn from the point P (1, 8) to the circle$${x^2}\, + \,{y^2}\, - \,6x\, - 4y\, - 11 = 0$$touch the circle at the points A and B. The e... A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. Th... A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. Th... A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. Th... Consider$$\,{L_1}:\,\,2x\,\, + \,\,3y\, + \,p\,\, - \,\,3 = 0\,{L_2}:\,\,2x\,\, + \,\,3y\, + \,p\,\, + \,\,3 = 0$$where p is a real number,... ABCD is a square of side length 2 units.$$C_1$$is the circle touching all the sides of the square ABCD and$$C_2$$is the circumcircle of square AB... ABCD is a square of side length 2 units.$$C_1$$is the circle touching all the sides of the square ABCD and$$C_2$$is the circumcircle of square AB... ABCD is a square of side length 2 units.$${C_1}$$is the circle touching all the sides of the square ABCD and$${C_2}$$is the circumcircle of squar... A circle is given by$${x^2}\, + \,{(y\, - \,1\,)^2}\, = \,1$$, another circle C touches it externally and also the x-axis, then thelocus of its cent... If one of the diameters of the circle$${x^2} + {y^2} - 2x - 6y + 6 = 0$$is a chord to the circle with centre (2, 1), then the radius of the circle i... The centre of circle inscibed in square formed by the lines$${x^2} - 8x + 12 = 0\,\,and\,{y^2} - 14y + 45 = 0$$, is If the tangent at the point P on the circle$${x^2} + {y^2} + 6x + 6y = 2$$meets a straight line 5x - 2y + 6 = 0 at a point Q on the y-axis, then th... Let A B be a chord of the circle$${x^2} + {y^2} = {r^2}$$subtending a right angle at the centre. Then the locus of the centriod of the triangle PAB ... Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of t... The triangle PQR is inscribed in the circle$${x^2}\, + \,\,{y^2} = \,25$$. If Q and R have co-ordinates (3, 4) and ( - 4, 3) respectively, then$$\a... If the circles $${x^2}\, + \,{y^2}\, + \,\,2x\, + \,2\,k\,y\,\, + \,6\,\, = \,\,0,\,\,{x^2}\, + \,\,{y^2}\, + \,2ky\, + \,k\, = \,0$$ intersect orthog... 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 th... The angle between a pair of tangents drawn from a point P to the circle $${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\, + \,9\,{\sin ^2}\,\alpha \, + \,13... The circles$${x^2} + {y^2} - 10x + 16 = 0$$and$${x^2} + {y^2} = {r^2}$$intersect each other in two distinct points if The locus of the centre of a circle, which touches externally the circle$${x^2} + {y^2} - 6x - 6y + 14 = 0$$and also touches the y-axis, is given by... The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle$${x^2} + {y^2} = 9$$is If the two circles$${(x - 1)^2} + {(y - 3)^2} = {r^2}$$and$${x^2} + {y^2} - 8x + 2y + 8 = 0$$intersect in two distinct points, then The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle of area 154 sq. units. Then the equation of this circle is If a circle passes through the point (a, b) and cuts the circle$${x^2}\, + \,{y^2}\, = \,{k^2}$$orthogonally, then the equation of the locus of its ... The locus of the mid-point of a chord of the circle$${x^2} + {y^2} = 4$$which subtends a right angle at the origin is The centre of the circle passing through the point (0, 1) and touching the curve$$\,y = {x^2}$$at (2, 4) is The equation of the circle passing through (1, 1) and the points of intersection of$${x^2} + {y^2} + 13x - 3y = 0$$and$$2{x^2} + 2{y^2} + 4x - 7y ... A square is inscribed in the circle $${x^2} + {y^2} - 2x + 4y + 3 = 0$$. Its sides are parallel to the coordinate axes. The one vertex of the square i... Two circles $${x^2} + {y^2} = 6$$ and $${x^2} + {y^2} - 6x + 8 = 0$$ are given. Then the equation of the circle through their points of intersection a... ## Subjective Circles with radii 3, 4 and 5 touch each other externally. It P is the point of intersection of tangents to these circles at their points of contact, ... Find the equation of circle touching the line 2x + 3y + 1 = 0 at (1, -1) and cutting orthogonally the circle having line segment joining (0, 3) and (-... 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 cir... 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 $$C_2$$ exter... 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 firs... 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 havin... $$C_1$$ 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 d... 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 poi... 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 $$\,\alpha \... Find the intervals of value of a for which the line y + x = 0 bisects two chords drawn from a point$$\left( {{{1\, + \,\sqrt 2 a} \over 2},\,{{1\, - ... Find the coordinates of the point at which the circles $${x^2}\, + \,{y^2} - \,4x - \,2y = - 4\,\,and\,\,{x^2}\, + \,{y^2} - \,12x - \,8y = - 36$$ t... 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 $${x^2}\, + \,{y^2} - \,... Let a circle be given by 2x (x - a) + y (2y - b) = 0,$$(a\, \ne \,0,\,\,b\, \ne 0)$$. Find the condition on a abd b if two chords, each bisected by t... Two circles, each of radius 5 units, touch each other at (1, 2). If the equation of their common tangent is 4x + 3y = 10, find the equation of the cir... A circle touches the line y = x at a point P such that OP =$${4\sqrt 2 \,}$$, where O is the origin. The circle contains the point (- 10, 2) in its i... If$$\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}\... 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 o... 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 axe... 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 equa... The abscissa of the two points A and B are the roots of the equation $${x^2}\, + \,2ax\, - {b^2} = 0$$ and their ordinates are the roots of the equati... Through a fixed point (h, k) secants are drawn to the circle $$\,{x^2}\, + \,{y^2} = \,{r^2}$$. Show that the locus of the mid-points of the secants i... Let A be the centre of the circle $${x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, - 20 = 0\,$$. Suppose that the tangents at the points B (1, 7) and D (4. - 2... Find the equations of the circle passing through (- 4, 3) and touching the lines x + y = 2 and x - y = 2. Find the equation of the circle whose radius is 5 and which touches the circle $${x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, - 20 = 0\,$$ at the point (5, 5... ## Fill in the Blanks 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 mov... 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........ 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......................... 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 i... 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 v... The area of the triangle formed by the positive x-axis and the normal and the tangent to the circle $${x^2} + {y^2} = 4\,\,at\,\,\left( {1,\sqrt 3 } \... If the circle$${C_1}:{x^2} + {y^2} = 16$$intersects another circle$${C_2}$$of radius 5 in such a manner that common chord is of maximum lenght and... The area of the triangle formed by the tangents from the point (4, 3) to the circle$${x^2} + {y^2} = 9$$and the line joining their points of contac... From the point A(0, 3) on the circle$${x^2} + 4x + {(y - 3)^2} = 0$$, a chord AB is drawn and extended to a point M such that AM = 2AB. The equation ... The equation of the line passing through the points of intersection of the circles$$3{x^2} + 3{y^2} - 2x + 12y - 9 = 0$$and$${x^2} + {y^2} - 6x + 2... 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............... 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 a quadrilateral of area......... The lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to the same circle. The radius of this circle is ........................................ The point of intersection of the line 4x - 3y - 10 = 0 and the circle $${x^2} + {y^2} - 2x + 4y - 20 = 0$$ are ........................and .............. 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 cannot be equal to ................... ## True or False The line x + 3y = 0 is a diameter of the circle $${x^2} + {y^2} - 6x + 2y = 0\,$$. No tangent can be drawn from the point (5/2, 1) to the circumcircle of the triangle with vertices$\$\left( {1,\sqrt 3 } \right)\,\,\left( {1, - \sqrt ...
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