Circle PYQs - Last 10 Years

$P$ and $Q$ are the points of trisection of the line segment joining the points $(3,-7)$ and $(-5,3)$. If $P Q$ subtends right angle at a variable point $R$, then the locus of $R$ is

If $A(1,2), B(2,1)$ are two vertices of an acute angled triangle and $S(0,0)$ is its circumcenter, then the angle subtended by $A B$ at the third vertex is

If the common chord of the circles $x^{2}+y^{2}-2 x+2 y+1=0$ and $x^{2}+y^{2}-2 x-2 y-2=0$ is the diameter of a circle $S$, then the center of the circles is

$x-2 y-6=0$ is a normal to the circle $x^{2}+y^{2}+2 g x+2 f y-8=0$. If the line $y=2$ touches this circle, then the radius of the circle can be

A circle passing through the points $(1,1)$ and $(2,0)$ touches the line $3 x-y-1=0$. If the equation of this circle is $x^{2}+y^{2}+2 g x+2 f y+c=0$, then a possible value of $g$ is

The line $x+y+1=0$ intersects the circle $x^{2}+y^{2}-4 x+2 y-4=0$ at the points $A$ and $B$. If $M(a, b)$ is the mid-point of $A B$, then $a-b=$

A circle passes through the points $(2,0)$ and $(1,2)$. If the power of the point $(0,2)$ with respect to this circle is 4 , then the radius of the circle is

A circle $S$ passes through the points of intersection of the circles $x^{2}+y^{2}-2 x-3=0$ and $x^{2}+y^{2}-2 y=0$. If $x+y+1=0$ is a tangent to the circle $S$, then equation of $S$ is

The slope of a common tangent to the circles $x^2+y^2-4 x-8 y+16=0$ and $x^2+y^2-6 x-16 y+64=0$ is

If $(p, q)$ is the centre of the circle which cuts the three circles $x^2+y^2-2 x-4 y+4=0, x^2+y^2+2 x-4 y+1=0$ and $x^2+y^2-4 x-2 y-11=0$ orthogonally, then $p+q=$

If $(h, k)$ is the internal centre of similitude of the circles $x^2+y^2+2 x-6 y+1=0$ and $x^2+y^2-4 x+2 y+4=0$, then $4 h=$

If the tangents $x+y+k=0$ and $x+a y+b=0$ drawn to the circle $S=x^2+y^2+2 x-2 y+1=0$ are perpendicular to each other and $k, b$ are both greater than 1 , then $b-k=$

$(1, k)$ is a point on the circle passing through the points $(-1,1),(0,-1)$ and $(1,0)$. If $k \neq 0$, then $k=$

$x^2+y^2+2 x-6 y-6=0$ and $x^2+y^2-6 x-2 y+k=0$ are two intersecting circles and $k$ is not an integer. If $\theta$ is the angle between the two circles and $\cos \theta=\frac{-5}{24}$, then $k=$

A rhombus is inscribed in the region common to the two circles $x^{2}+y^{2}-4 x-12=0$ and $x^{2}+y^{2}+4 x-12=0$. If the line joining the centres of these circles and the common chord of them are the diagonals of this rhombus, then the area...

A rectangle is formed by the lines $x=4, x=-2, y=5, y=-2$ and a circle is drawn through the vertices of this rectangle. The pole of the line $y+2=0$ with respect to this circle is

If the equation of the circle which cuts each of the circles $x^{2}+y^{2}=4, x^{2}+y^{2}-6 x-8 y+10=0$ and $x^{2}+y^{2}+2 x-4 y-2=0$ at the extremities of a diameter of these circles is $x^{2}+y^{2}+2 g x+2 f y+c=0$, then $g+f+c=$

If $m$ is the slope and $P(8, \beta)$ is the mid-point of a chord of contact of the circle $x^{2}+y^{2}=125$, then the number of values of $\beta$ such that $\beta$ and $m$ are integers is

The equation of a circle which passes through the points of intersection of the circles $2 x^{2}+2 y^{2}-2 x+6 y-3=0, x^{2}+y^{2}+4 x+2 y+1=0$ and whose centre lies on the common chord of these circles is

The equation of the circle passing through the origin and cutting the circles $x^{2}+y^{2}+6 x-15=0$ and $x^{2}+y^{2}-8 y-10=0$ orthogonally is

The radius of a circle touching all the four circles $(x \pm \lambda)^2+(y \pm \lambda)^2=\lambda^2$ is

If the angle between the pair of tangents drawn to the circle $x^2+y^2-2 x+4 y+3=0$ from the point $(6,-5)$ is $\theta$, then $\cot \theta=$

If the radical centre of the given three circles $x^2+y^2=1, x^2+y^2-2 x-3=0$ and $x^2+y^2-2 y-3=0$ is $C(\alpha, \beta)$ and $r$ is the sum of the radii of the given circles, then the circle with $C(\alpha, \beta)$ as centre and $r$ as rad...

If the angle between the circles $x^2+y^2-4 x-6 y+k=0$ and $x^2+y^2+8 x-4 y+11=0$ is $\frac{\pi}{2}$, then the value of $k$ is

If a point $P$ moves so that the distance from $(0,2)$ to $P$ is $\frac{1}{\sqrt{2}}$ times the distance of $P$ from $(-1,0)$, then the locus of the point $P$ is

If the parametric equations of the circle passing through the points $(3,4),(3,2)$ and $(1,4)$ is $x=a+r \cos \theta, y=b+r \sin \theta$, then $b^a r^a=$

A tangent $P T$ is drawn to the circle $x^2+y^2=4$ at the point $P(\sqrt{3}, 1)$. If a straight line $L$ which is perpendicular to $P T$ is a tangent to the circle $(x-3)^2+y^2=1$, then a possible equation of $L$ is

The area (in sq units) of the triangle formed by the $x$-axis, the tangent and the normal drawn to the circle $x^2+y^2=10 x$ at the point $(9,3)$ is

If the circumcenter of the triangle formed by the points $A(a, 3), B(b, 5)$ and $C(a, b)$ is $(1,1)$, then out of all the possible coordinates of $C$ the sum of the absolute values of the distinct coordinates of $C$ is

The equation of a circle passing through $(-6,3)$ and touching both the coordinates axes is

If the equation of the circle whose radius is $\sqrt{10}$ and which touches the circle $x^2+y^2+2 x+8 y-23=0$ externally at the point $(1,2)$ is $x^2+y^2+a x+b y+c=0$, then $|a+b+c|=$

The number of common tangents of the circles $x^2+y^2-4=0$ and $x^2+y^2-6 x-8 y-24=0$ is

The equation of the circle passing through the points of intersection of the circles $x^2+y^2+6 x+4 y-12=0$, $x^2+y^2-4 x-6 y-12=0$ and having radius $\sqrt{13}$ is

If a circle ' $S$ ' passing through the origin and having its centre on the line $x-y=0$ cuts the circle $x^2+y^2-4 x-6 y+10=0$ orthogonally, then the diameter of ' $S$ ' is

Let a chord $A B$ subtend an angle of $60^{\circ}$ at the centre $C(2,3)$ of a circle $S$. If the equation of $A B$ is $x+y+1=0$, then the equation of the circle $S$ is

If the angle between the circles $x^2+y^2-2 x+2 y+1=0$ and $x^2+y^2+2 x-2 y+k=0$ is $\frac{\pi}{3}$, then

Let 6,8 be the $X$ and $Y$-intercepts made by the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$, respectively. If $g x+f y+1=0$ is a line passing through the point $(1,-1)$, then the radius of the circle $S=0$ is

If $(3,1)$ and $(-2,4)$ are points on a circle $S$ whose centre lies on the line $x-y+1=0$, then the parametric equations of $S$ are

Let the line $x-y+1=0$ intersect the circle $x^2+y^2+2 x+2 y+1=0$ in two points $A$ and $B$. If $A B$ is the diameter of the circle $x^2+y^2+2 g x+2 f y+c=0$, then $g+f=$

Let $S \equiv x^2+y^2-8 x+10 y+5=0$ be a circle. Let $P(1,1)$ and $Q(1,-1)$ be two points. Then, the point of intersection of the polar of $P$ with respect to $S=0$ and the chord with $Q$ as mid-point to $S=0$ is

If the equation $2 x-3 y+3=0,2 x+y+1=0$ and $6 x+4 y+1=0$ represent the sides of a triangle, then the equation of the circle passing through the vertices of this triangle is

The equation of the circle inscribed in a square formed by the lines $x+y-2=0, x+y-6=0, x-y+1=0$ and $x-y+5=0$ is

Let the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$ touch the positive $X$-axis and the positive $Y$-axis. Let $(2,4)$ be a point on the circle $S=0$. If two such circles exist, then the difference of their areas is

If the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$ cuts each of the three circles $x^2+y^2+4 x+4 y+7=0$, $x^2+y^2-4 x+4 y+7=0$ and $x^2+y^2-4 x-4 y+7=0$ orthogonally, then the equation of the tangent drawn at the point $(\sqrt{3}, 2)$ to the ...

If $T_1 T^{\prime}{ }_1$ and $T_2 T_2^{\prime}$ are the common tangents of the circles $S \equiv x^2+y^2-2 x-4 y-4=0$ and $S \equiv x^2+y^2+4 x+4=0$, where $T_1, T^{\prime}{ }_1, T_2, T^{\prime}{ }_2$ are the points of contact, then the dis...

A circle $S \equiv x^2+y^2+2 g x+2 f y+4=0$ cuts the circle $x^2+y^2-4 x-4 y-4=0$ orthogonally and makes an angle of $60^{\circ}$ with the circle $x^2+y^2+4 x+4 y+4=0$. Then, the radius of the circle $S=0$ is

The image of every point lying on the curve $x^2+y^2=1$ in the line $x+y=1$ satisfies the equation

If $C(\alpha, \beta)(a<0)$ is the centre of the circle that touches the $Y$-axis at $(0,3)$ and makes an intercept of length 2 units on positive $X$-axis, then $(\alpha, \beta)=$

If a circle passing through $A(1,1)$ touches the $X$-axis, then the locus of the other end of the diameter through $A$ is

If a diameter of the circle $x^2+y^2-4 x+6 y-12=0$ is a chord of a circle $S$ whose centre is at $(-3,2)$, then the radius of $S$ is