Circle PYQs - Last 10 Years

Let $P$ be the point on the parabola $y = x^2$ such that the slope of the tangent to the parabola at the point $P$ is $4$. Let $Q$ be the point in the first quadrant lying on the circle $x^2 + y^2 = 2$ such that the slope of the tangent to ...

Let the straight line $y=2 x$ touch a circle with center $(0, \alpha), \alpha>0$, and radius $r$ at a point $A_1$. Let $B_1$ be the point on the circle such that the line segment $A_1 B_1$ is a diameter of the circle. Let $\alpha+r=5+\sqrt{...

Let $C_1$ be the circle of radius 1 with center at the origin. Let $C_2$ be the circle of radius $r$ with center at the point $A=(4,1)$, where $1 < r < 3$. Two distinct common tangents $P Q$ and $S T$ of $C_1$ and $C_2$ are drawn. The tange...

Let $A_1, A_2, A_3, \ldots, A_8$ be the vertices of a regular octagon that lie on a circle of radius 2 . Let $P$ be a point on the circle and let $P A_i$ denote the distance between the points $P$ and $A_i$ for $i=1,2, \ldots, 8$. If $P$ va...

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 $G_{1}, G_{2}, \ldots, G_{n}$ touches the circle $G$ externally. Also, for $i=1,2, \ldots, n-1$, th...

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 \(A B, A C\) and also touches internally the circumcircle of the triangle \(A B C\), then the value of $...

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 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 circles among these k circles such that no two circles intersect. Then

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 in R and have centers on the x-axis. Let C be the circle that has largest radius among the...

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 in R and have centers on the x-axis. Let C be the circle that has largest radius among the...

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 equation of the circle passing through the vertices of the triangle \(\Delta\) is

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 line passing through P and Q is 2x + 4y = 5. If the centre of the circumcircle of the triangle ...

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-coordinate \(- {3 \over 5}\), then which one of the following options is correct?

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 circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the ci...

Let S be the circle in the XY-plane defined the equation x2 + y2 = 4.Let E1E2 and F1F2 be the chords of S passing through the point P0 (1, 1) and parallel to the X-axis and the Y-axis, respectively. Let G1G2 be the chord of S passing throug...