Circle PYQs - Last 10 Years

Consider the circle C : $x^2+y^2-6 x-8 y-11=0$. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle $x^2+y^2-\alpha x...

Let the centre of the circle $x^2+y^2+2 \mathrm{~g} x+2 f y+25=0$ be in the first quadrant and lie on the line $2 x-y=4$. Let the area of an equilateral triangle inscribed in the circle be $27 \sqrt{3}$. Then the square of the length of the...

Let the line $x-y=4$ intersect the circle $\mathrm{C}:(x-4)^2+(y+3)^2=9$ at the points Q and R . If $\mathrm{P}(\alpha, \beta)$ is a point on C such that $\mathrm{PQ}=\mathrm{PR}$, then $(6 \alpha+8 \beta)^2$ is equal to $\_\_\_\_$ .

Let C be a circle having centre in the first quadrant and touching the $x$-axis at a distance of 3 units from the origin. If the circle $C$ has an intercept of length $6 \sqrt{3}$ on $y$-axis, then the length of the chord of the circle C on...

Let P be a moving point on the circle $x^2+y^2-6 x-8 y+21=0$. Then, the maximum distance of P from the vertex of the parabola $x^2+6 x+y+13=0$ is equal to:

Let the point P be the vertex of the parabola $y=x^2-6 x+12$. If a line passing through the point P intersects the circle $x^2+y^2-2 x-4 y+3=0$ at the points R and S , then the maximum value of $(\mathrm{PR}+\mathrm{PS})^2$ is :

Suppose that two chords, drawn from the point $(1,2)$ on the circle $x^2+y^2+x-3 y=0$ are bisected by the $y$-axis. If the other ends of these chords are R and S , and the mid point of the line segment RS is $(\alpha, \beta)$, then $6(\alph...

Let a circle C have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of C on the line $x + y = 1$ is $\sqrt{14}$, then ...

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines $x + (k-1)y + 3 = 0$ and $2x + k^2y - 4 = 0$. If the line $x - y + 2 = 0$ intersects the circle at the points A and B, then...

Let $y=x$ be the equation of a chord of the circle $\mathrm{C}_1$ (in the closed half-plane $x \geq 0$ ) of diameter 10 passing through the origin. Let $\mathrm{C}_2$ be another circle described on the given chord as its diameter. If the eq...

Let the circle $x^2 + y^2 = 4$ intersect x-axis at the points A$(a, 0)$, $a > 0$ and B$(b, 0)$. Let $P(2 \cos \alpha, 2 \sin \alpha)$, $0 < \alpha < \frac{\pi}{2}$ and $Q(2 \cos \beta, 2 \sin \beta)$ be two points such that $(\alpha - \beta...

Let a circle of radius 4 pass through the origin O , the points $\mathrm{A}(-\sqrt{3} a, 0)$ and $\mathrm{B}(0,-\sqrt{2} b)$, where $a$ and $b$ are real parameters and $a b \neq 0$. Then the locus of the centroid of $\triangle \mathrm{OAB}$...

Let the set of all values of $r$, for which the circles $(x+1)^2+(y+4)^2=r^2$ and $x^2+y^2-4 x-2 y-4=0$ intersect at two distinct points be the interval $(\alpha, \beta)$. Then $\alpha \beta$ is equal to

Let PQ and MN be two straight lines touching the circle $x^2+y^2-4 x-6 y-3=0$ at the points $A$ and $B$ respectively. Let $O$ be the centre of the circle and $\angle A O B=\pi / 3$. Then the locus of the point of intersection of the lines P...

If $P$ is a point on the circle $x^2+y^2=4, Q$ is a point on the straight line $5 x+y+2=0$ and $x-y+1=0$ is the perpendicular bisector of PQ , then 13 times the sum of abscissa of all such points P is $\_\_\_\_$ .

Let $C_1$ be the circle in the third quadrant of radius 3 , that touches both coordinate axes. Let $C_2$ be the circle with centre $(1,3)$ that touches $\mathrm{C}_1$ externally at the point $(\alpha, \beta)$. If $(\beta-\alpha)^2=\frac{m}{...

Let $C$ be the circle $x^2+(y-1)^2=2, E_1$ and $E_2$ be two ellipses whose centres lie at the origin and major axes lie on x -axis and y -axis respectively. Let the straight line $x+y=3$ touch the curves $C, E_1$ and $E_2$ at $P\left(x_1, y...

If the four distinct points $(4,6),(-1,5),(0,0)$ and $(k, 3 k)$ lie on a circle of radius $r$, then $10 k+r^2$ is equal to

The absolute difference between the squares of the radii of the two circles passing through the point $(-9,4)$ and touching the lines $x+y=3$ and $x-y=3$, is equal to ________ .

Let the line x+y=1 meet the circle $x^2+y^2=4$ at the points A and B. If the line perpendicular to AB and passing through the mid-point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ABCD is equal to :

Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on 3x + 2y + 2 = 0. Then the length of the chord, of the circle C, whose mid-point is (1, 2), is:

Let the equation of the circle, which touches $x$-axis at the point $(a, 0), a>0$ and cuts off an intercept of length $b$ on $y-a x i s$ be $x^2+y^2-\alpha x+\beta y+\gamma=0$. If the circle lies below $x-a x i s$, then the ordered pair $\l...

Let circle $C$ be the image of $x^2+y^2-2 x+4 y-4=0$ in the line $2 x-3 y+5=0$ and $A$ be the point on $C$ such that $O A$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\b...

Let the circle $C$ touch the line $x-y+1=0$, have the centre on the positive $x$-axis, and cut off a chord of length $\frac{4}{\sqrt{13}}$ along the line $-3 x+2 y=1$. Let H be the hyperbola $\frac{x^2}{\alpha^2}-\frac{y^2}{\beta^2}=1$, who...

A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point $(2,5)$ and intersects the circle $C$ at exactly two points. If the set of all possible va...

Let the centre of a circle, passing through the points \((0,0),(1,0)\) and touching the circle \(x^2+y^2=9\), be \((h, k)\). Then for all possible values of the coordinates of the centre \((h, k), 4\left(h^2+k^2\right)\) is equal to _______...

Let a circle passing through \((2,0)\) have its centre at the point \((\mathrm{h}, \mathrm{k})\). Let \((x_{\mathrm{c}}, y_{\mathrm{c}})\) be the point of intersection of the lines \(3 x+5 y=1\) and \((2+\mathrm{c}) x+5 \mathrm{c}^2 y=1\). ...

Let the circles \(C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2\) and \(C_2:(x-8)^2+\left(y-\frac{15}{2}\right)^2=r_2^2\) touch each other externally at the point \((6,6)\). If the point \((6,6)\) divides the line segment joining the centres of the ci...

If the image of the point \((-4,5)\) in the line \(x+2 y=2\) lies on the circle \((x+4)^2+(y-3)^2=r^2\), then \(r\) is equal to:

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are \(m\) and \(n\), respectively, then \(m+n^2\) is equal to

If \(\mathrm{P}(6,1)\) be the orthocentre of the triangle whose vertices are \(\mathrm{A}(5,-2), \mathrm{B}(8,3)\) and \(\mathrm{C}(\mathrm{h}, \mathrm{k})\), then the point \(\mathrm{C}\) lies on the circle :

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point \((3,2)\) and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point \((5,5)\) is :

Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius r of the circle passing through the point F and touching the line segments BC and...

Let the circle \(C_1: x^2+y^2-2(x+y)+1=0\) and \(\mathrm{C_2}\) be a circle having centre at \((-1,0)\) and radius 2 . If the line of the common chord of \(\mathrm{C}_1\) and \(\mathrm{C}_2\) intersects the \(\mathrm{y}\)-axis at the point ...

A square is inscribed in the circle \(x^2+y^2-10 x-6 y+30=0\). One side of this square is parallel to \(y=x+3\). If \(\left(x_i, y_i\right)\) are the vertices of the square, then \(\Sigma\left(x_i^2+y_i^2\right)\) is equal to:

Let \(\mathrm{C}\) be a circle with radius \(\sqrt{10}\) units and centre at the origin. Let the line \(x+y=2\) intersects the circle \(\mathrm{C}\) at the points \(\mathrm{P}\) and \(\mathrm{Q}\). Let \(\mathrm{MN}\) be a chord of $$\mathr...

If one of the diameters of the circle \(x^2+y^2-10 x+4 y+13=0\) is a chord of another circle \(\mathrm{C}\), whose center is the point of intersection of the lines \(2 x+3 y=12\) and \(3 x-2 y=5\), then the radius of the circle $$\mathrm{C}...

Let a variable line passing through the centre of the circle \(x^2+y^2-16 x-4 y=0\), meet the positive co-ordinate axes at the points \(A\) and \(B\). Then the minimum value of \(O A+O B\), where \(O\) is the origin, is equal to

If the circles \((x+1)^2+(y+2)^2=r^2\) and \(x^2+y^2-4 x-4 y+4=0\) intersect at exactly two distinct points, then

Consider two circles \(C_1: x^2+y^2=25\) and \(C_2:(x-\alpha)^2+y^2=16\), where \(\alpha \in(5,9)\). Let the angle between the two radii (one to each circle) drawn from one of the intersection points of \(C_1\) and \(C_2\) be $$\sin ^{-1}\l...

Equations of two diameters of a circle are \(2 x-3 y=5\) and \(3 x-4 y=7\). The line joining the points \(\left(-\frac{22}{7},-4\right)\) and \(\left(-\frac{1}{7}, 3\right)\) intersects the circle at only one point \(P(\alpha, \beta)\). The...

Four distinct points $(2 k, 3 k),(1,0),(0,1)$ and $(0,0)$ lie on a circle for $k$ equal to :

Consider a circle \((x-\alpha)^2+(y-\beta)^2=50\), where \(\alpha, \beta>0\). If the circle touches the line \(y+x=0\) at the point \(P\), whose distance from the origin is \(4 \sqrt{2}\), then \((\alpha+\beta)^2\) is equal to __________.

Let $C: x^2+y^2=4$ and $C^{\prime}: x^2+y^2-4 \lambda x+9=0$ be two circles. If the set of all values of $\lambda$ so that the circles $\mathrm{C}$ and $\mathrm{C}$ intersect at two distinct points, is $\mathrm{R}-[\mathrm{a}, \mathrm{b}]$,...

Let the locus of the midpoints of the chords of the circle $x^2+(y-1)^2=1$ drawn from the origin intersect the line $x+y=1$ at $\mathrm{P}$ and $\mathrm{Q}$. Then, the length of $\mathrm{PQ}$ is :

Consider a circle \(C_{1}: x^{2}+y^{2}-4 x-2 y=\alpha-5\). Let its mirror image in the line \(y=2 x+1\) be another circle \(C_{2}: 5 x^{2}+5 y^{2}-10 f x-10 g y+36=0\). Let \(r\) be the radius of \(C_{2}\). Then \(\alpha+r\) is equal to ___...

Let O be the origin and OP and OQ be the tangents to the circle \(x^2+y^2-6x+4y+8=0\) at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point \(\left( {\alpha ,{1 \over 2}} \right)\), then a value of $$...

A circle passing through the point \(P(\alpha, \beta)\) in the first quadrant touches the two coordinate axes at the points \(A\) and \(B\). The point \(P\) is above the line \(A B\). The point \(Q\) on the line segment \(A B\) is the foot...

Let the point \((p, p+1)\) lie inside the region \(E=\left\{(x, y): 3-x \leq y \leq \sqrt{9-x^{2}}, 0 \leq x \leq 3\right\}\). If the set of all values of \(\mathrm{p}\) is the interval \((a, b)\), then \(b^{2}+b-a^{2}\) is equal to _______...

If the tangents at the points \(\mathrm{P}\) and \(\mathrm{Q}\) on the circle \(x^{2}+y^{2}-2 x+y=5\) meet at the point \(R\left(\frac{9}{4}, 2\right)\), then the area of the triangle \(\mathrm{PQR}\) is :