Circle

If the circle $S=0$ cuts the circles $x^2+y^2-2 x+6 y=0$, $x^2+y^2-4 x-2 y+6=0$ and $x^2+y^2-12 x+2 y+3=0$ orthogonally, then equation of the tangent at $(0,3)$ on $S=0$ is

If $\theta$ is the angle between the tangents drawn from the point $(2,3)$ to the circle $x^2+y^2-6 x+4 y+12=0$ then $\theta=$

A circle $s$ passes through the points of intersection of the circles $x^2+y^2-2 x+2 y-2=0$ and $x^2+y^2+2 x-2 y+1=0$. If the centre of this circle $S$ lies on the line $x-y+6=0$, then the radius of the circle $S$ is

If $(a, b)$ and ( $c, d)$ are the internal and external centres of similitudes of the circles $x^2+y^2+4 x-5=0$ and $x^2+y^2-6 y+8=0$ respectively, then $(a+d)(b+q)=$

If $Q(h, k)$ is the inverse point of the point $P(1,2)$ with respect to the circle $x^2+y^2-4 x+1=0$, then $2 h+k=$

If $2 x-3 y+3=0$ and $x+2 y+k=0$ are conjugate lines with respect to the circle $S=x^2+y^2+8 x-6 y-24=0$, then the length of the tangent drawn from the point $\left(\frac{k}{4}, \frac{k}{3}\right)$ to the circle $S=0$, is

Let $S$ be the circumcircle of the triangle formed by the line $x-2 y-4=0$ with the coordinate axes. If $P(-2,-4)$ is a point in the plane of the circle $S$ and $Q$ is a point on $S$ such that the distance between $P$ and $Q$ is the least, ...

If the chord of contact of the point $P(h, k)$ with respect to the circle $x^2+y^2-4 x-4 y+8=0$ meets the circle in two distinct points and it also makes an angle $45^{\circ}$ with the positive $X$-axis in the positive direction, then $(h, ...

The locus of a point which is at a distance of 2 units from the line $2 x-3 y+4=0$ and at a distance of $\sqrt{13}$ units from a point $(5,0)$, is

The equation of the pair of tangents drawn from the point $(1,1)$ to the circle $x^2+y^2+2 x+2 y+1=0$ is

Let the circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$ cut the circles $x^2+y^2-2 x+2 y-2=0$ and $x^2+y^2+4 x-6 y+9=0$ orthogonally. If the centre of the circle $S=0$ lies on the line $2 x+3 y-2=0$, then $2 g+f=$

If the coordinates of point of contact of the circles $x^2+y^2-4 x+8 y+4=0$ and $x^2+y^2+2 x=0$ is $(a, b)$, then $a+2 b=$

Let $P$ and $Q$ be the inverse points with respect to the circle $S \equiv x^2+y^2-4 x-6 y+k=0$ and $C$ be the centre of the circle $S=0$ such that $C P \cdot C Q=4$. If $P=(1,2)$ and $Q=(a, b)$, then $2 a=$

Let $A(2,3), B(3,-1)$ and $C(-3,2)$ be three points. If the centre of the circle passing through $A, B$ and $C$ is $(h, k)$, then $2 k-4 h=$

If $P\left(\frac{\pi}{3}\right)$ and $Q\left(\frac{2 \pi}{3}\right)$ represent two points on the circle $x^2+y^2-4 x+6 y-12=0$ in parametric form, then the length of the chord $P Q$ is

The distance between the centres of similitude of the circles $x^2+y^2+6 x-8 y+16=0$ and $x^2+y^2-2 x-2 y+1=0$ is

Let $A(1,2)$ be the centre and 3 be the radius of a circle $S$. Let $B(-1,-1)$ be the centre and $r$ be the radius of another circle $S^{\prime}$. If $\frac{\pi}{3}$ is the angle between the circles $S$ and $S^{\prime}$, then the number of ...

The radical centre of the three circles \(x^2+y^2-1=0, x^2+y^2-8 x+15=0\) and \(x^2+y^2+10 y+24=0\) is

Suppose that the \(x\)-coordinates of the points \(A\) and \(B\) satisfy \(x^2+2 x-a^2=0\) and their \(y\)-coordinates satisfy \(y^2+4 y-b^2=0\). Then, the equation of the circle with \(A B\) as its diameter is

The least distance of the point \((10,7)\) from the circle \(x^2+y^2-4 x-2 y-20=0\) is

A circle is such that \((x-2) \cos \theta+(y-2) \sin \theta=1\) touches it for all values of \(\theta\). Then, the circle is

The radius of the circle having. \(3 x-4 y+4=0\) and \(6 x-8 y-7=0\) as its tangents is

The locus of mid-points of points of intersection of \(x \cos \theta+y \sin \theta=1\) with the coordinate axes is

If the tangent at the point \(P\) on the circle \(x^2+y^2+6 x+6 y=2\) meets the straight line \(5 x-2 y+6=0\) at a point \(Q\) on the \(Y\)-axis, then the length of \(P Q\) is

For any two non-zero real numbers \(a\) and \(b\) if this line \(\frac{x}{a}+\frac{y}{b}=1\) is a tangent to the circle \(x^2+y^2=1\), then which of the following is true?

The locus of centers of the circles, possessing the same area and having \(3 x-4 y+4=0\) and \(6 x-8 y-7=0\) as their common tangent, is

The pole of the line \(\frac{x}{a}+\frac{y}{b}=1\) with respect to the circle \(x^2+y^2=c^2\) is

The length of the intercept on the line \(4 x-3 y-10=0\) by the circle \(x^2+y^2-2 x+4 y-20=0\) is

The ratio of the largest and shortest distances from the point \((2,-7)\) to the circle \(x^2+y^2-14 x-10 y-151=0\) is

The area of the circle passing through the points \((5, \pm 2),(1,2)\) is

The image of the point \((3,4)\) with respect to the radical axis of the circles \(x^2+y^2+8 x+2 y+10=0\) and \(x^2+y^2+7 x+3 y+10=0\) is

A circle has its centre in the first quadrant and passes through \((2,3)\). If this circle makes intercepts of length 3 and 4 respectively on \(x=2\) and \(y=3\), its equation is

For any real number \(t\), the point \(\left(\frac{8 t}{1+t^2}, \frac{4\left(1-t^2\right)}{1+t^2}\right)\) lies on a / an

Find the equation of the circle which passes through the point \((1,2)\) and the points of intersection of the circles \(x^2+y^2-8 x-6 y+21=0\) and \(x^2+y^2-2 x-15=0\)

The length of the common chord of the circles \(x^2+y^2+3x+5y+4=0\) and \(x^2+y^2+5x+3y+4=0\) is __________ units.

Let \(C\) be the circle center \((0,0)\) and radius 3 units. The equation of the locus of the mid-points of the chords of the circle \(c\) that subtends an angle of \(\frac{2 \pi}{3}\) at its centre is

If the circle \(x^2+y^2-4 x-8 y-5=0\) intersects the line \(3 x-4 y-m=0\) in two distinct points, then the number of integral values of '\(m\)' is

The angle between the pair of tangents drawn from \((1,1)\) to the circle \(x^2+y^2+4 x+4 y-1=0\) is

Find the equation of the circle which passes through origin and cuts off the intercepts \(-\)2 and 3 over the \(X\) and \(Y\)-axes respectively.

The locus of a point, which is at a distance of 4 units from \((3,-2)\) in \(x y\)-plane is

The radical axis of the circles \(S_1: x^2+y^2-4 x+6 y-10=0\) and \(S_2 : x^2+y^2+2 x-6 y+2=0\), cut the circle \(S_1\) in

The equation of radical axis of the circles \(x^2+y^2+4 x+6 y+7=0\) and \(4 x^2+4 y^2+8 x+12 y-9=0\) is

If the power of the point \((1,6)\) with respect to the circle \(x^2+y^2+4 x-6 y-a=0\) is \(-16\) then \(a\) equals

The poles of the tangents to the circle \(x^2+y^2=4\) with respect to the circle \((x+2)^2+y^2=8\), lie on

The center and radius of the circle \(x^2+y^2+8 x+10 y-8=0\) respectively are and units

The points where the circle \(x^2+y^2-3 x -4 y+2=0\) cuts the \(X\)-axis are

The equation of the pair of straight lines parallel to \(x\)-axis and touching the circle \(x^2+y^2-6 x-4 y-12=0\) is

The point which has the same power with respect to each of the circles \(x^2+y^2-8 x+40=0, x^2+y^2-5 x+16=0\) and \(x^2+y^2-8 x+16 y+160=0\) is

The radius of the circle whose center lies at \((1,2)\) while cutting the circle \(x^2+y^2+4 x+16 y-30=0\) orthogonally, is units.

Let \(L_1\) be a straight line passing through the origin and \(L_2\) be the straight line \(x+y=1\). If the intercepts made by the circle \(x^2+y^2-x+3 y=0\) on \(L_1\) and \(L_2\) are equal, then which of the following equations represent...