The circles $x^2+y^2-2 x-4 y-4=0$ and $x^2+y^2+2 x+4 y-11=0$

If the line $4 x-3 y+7=0$ touches the circle $x^2+y^2-6 x+4 y-12=0$ at $(\alpha, \beta)$, then $\alpha+2 \beta=$

The slope of the common tangent drawn to the circles $x^2+y^2-4 x+12 y-216=0$ and $x^2+y^2+6 x-12 y+36=0$ is

If $r_1$ and $r_2$ are radii of two circles touching all the four circles $(x \pm r)^2+(y \pm r)^2=r^2$, then $\frac{r_1+r_2}{r}=$

If the equation of the circle having the common chord to the circles $x^2+y^2+x-3 y-10=0$ and $x^2+y^2+2 x-y-20=0$ as its diameter is $x^2+y^2+\alpha x+\beta y+\gamma=0$, then $\alpha+2 \beta+\gamma=$

The locus of the third vertex of a right-angled triangle, the ends of whose hypotenuse are $(1,2)$ and $(4,5)$ is

A circle touches both the coordinate axes and the straight line $L \equiv 4 x+3 y-6=0$ in the first quadrant. If this circle lies below the line $L=0$, then the equation of that circle is

If the smallest circle through the points of intersection of $x^2+y^2=a^2$ and $x \cos \alpha+y \sin \alpha=p, 0

If the lines $3 x-4 y+4=0$ and $6 x-8 y-7=0$ are the tangents to the same circle, then the area of that circle (in sq. units) is

Circles are drawn through the point $(2,0)$ to cut intercepts of length 5 units on the $X$-axis. If their centre lie in the first quadrant, then their equation is

If $A(\cos \alpha, \sin \alpha), B(\sin \alpha,-\cos \alpha), C(1,2)$ are the vertices of a $\triangle A B C$, then the locus of its centroid is

A circle passing through origin cuts the coordinate axes is $A$ and $B$. If the straight line $A B$ passes through a fixed point $\left(x_1, y_1\right)$, then the locus of the centre of the circle is

If $(\alpha, \beta)$ is the external centre of similitude of the circles $x^2+y^2=3$ and $x^2+y^2-2 x+4 y+4=0$, then $\frac{\beta}{\alpha}=$

The equation of the circle touching the lines $|x-2|+|y-3|=4$ is

If the chord joining the points $(1,2)$ and $(2,-1)$ on a circle subtends an angle of $\frac{\pi}{4}$ at any point on its circumference, then the equation of such a circle is

The equation of the circle which cuts all the three circles $4(x-1)^2+4(y-1)^2=1,4(x+1)^2+4(y-1)^2$ and $4(x+1)^2+4(y+1)^2=1$ orthogonally is

$A(a, 0)$ is a fixed point and $\theta$ is a parameter such that $0<\theta<2 \pi$. If $P(a \cos \theta, a \sin \theta)$ is a point on the circle $x^2+y^2=a^2$ and $Q(b \sin \theta,-b \cos \theta)$ is a point on the circle $x^2+y^2=b^2$, then the locus of the centroid of the $\triangle A P Q$ is

If the equation of the circle passing through the point $(8,8)$ and having the lines $x+2 y-2=0$ and $2 x+3 y-1=0$ as its diameters is $x^2+y^2+p x+q y+r=0$, then $p^2+q^2+r=$

If $2 x-3 y+1=0$ is the equation of the polar of a point $P\left(x_1, y_1\right)$ with respect to the circle $x^2+y^2-2 x+4 y+3=0$, then $3 x_1-y_1=$

If a unit circle $S \equiv x^2+y^2+2 g x+2 f y+c=0$ touches the circle $S^{\prime} \equiv x^2+y^2-6 x+6 y+2=0$ externally at the point $(-1,-3)$, then $g+f+c=$

$3 x+4 y-43=0$ is a tangent to the circle $S \equiv x^2+y^2-6 x+8 y+k=0$ at a point $P$. If $C$ is the centre of the circle and $Q$ is a point which divides $C P$ in the ratio $-1: 2$, then the power of the point $Q$ with respect to the circle $S=0$ is

If the radical axis of the circles $x^2+y^2+2 g x+2 f y+c=0$ and $2 x^2+2 y^2+3 x+8 y+2 c=0$ touches the circle $x^2+y^2+2 x+2 y+1=0$, then

After the coordinate axes are rotated through an angle $\frac{\pi}{4}$ in the anti-clockwise direction without shifting the origin, if the equation $x^2+y^2-2 x-4 y-20=0$ transforms to $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ in the new coordinate system, then

$ \left|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right|= $

If the circles $x^2+y^2+5 k x+2 y+k=0$ and $2 x^2+2 y^2+2 k x+3 y-1=0, k \in R$ intersect at points $P$ and $Q$ then the line $4 x+5 y-k=0$ passes through $P$ and $Q$ for

The slope of one of the direct common tangents drawn to the circles $x^2+y^2-2 x+4 y+1=0$ and $x^2+y^2-4 x-2 y+4=0$ is

If the pole of the line $x+2 b y-5=0$ with respect to the circle $S \equiv x^2+y^2-4 x-6 y+4=0$ lies on the line $x+b y+1=0$, then the polar of the point $(b,-b)$ with respect to the circle $S=0$ is

If $P(\alpha, \beta)$ is the radical centre of the circles $S \equiv x^2+y^2+4 x+7=0, S^{\prime}=2 x^2+2 y^2+3 x+5 y+9=0$ and $S^{\prime \prime} \equiv x^2+y^2+y=0$, then the length of the tangent drawn from $P$ to $S^{\prime}=0$ is

When the axes are rotated through an angle $\theta$ about origin in anti-clockwise direction and then translated to the new origin $(2,-2)$, if the transformed equation the equation of $x^2+y^2=4$ is $X^2+Y^2+a X+b Y+c=0$ then $a+b+c=$

From a point $P(-4,0)$, two tangents are drawn to the circle $x^2+y^2-4 x-6 y-12=0$ touching the circle at $A$ and $B$. If the equation of the circle passing through $P, A$ and $B$ is $x^2+y^2+2 g x+2 f y+c=0$, then $(g, f)=$

If the equation of the polar of the point $(\alpha,-1)$ with respect to the circle $x^2+y^2-4 x-6 y-12=0$ is $y=\beta$, then $4(\alpha+\beta)=$

If $\theta$ is the angle between the tangents drawn from the point $(-1,-1)$ to the circle $x^2+y^2-4 x-6 y+c=0$ and $\cos \theta=-\frac{7}{25}$, then the radius of the circle 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 $a=$

The radius of the circle passing through the points of intersection of the circles $x^2+y^2+2 x+4 y+1=0$, $x^2+y^2-2 x-4 y-4=0$ and intersecting the circle $x^2+y^2=6$ orthogonally is

A circle passing through the point $(1,0)$ makes an intercept of length 4 units on $X$-axis and an intercept of length $2 \sqrt{11}$ units on $Y$-axis. If the centre of the circle lies in the fourth quadrant, then the radius of the circle is

If $\left(\frac{1}{10}, \frac{-1}{5}\right)$ is the inverse point of a point $(-1,2)$ with respect to the circle $x^2+y^2-2 x+4 y+c=0$ then $c=$

If the equation of the circle lying in the first quadrant, touching both the coordinate axes and the line $\frac{x}{3}+\frac{y}{4}=1$ is $(x-c)^2+(y-c)^2=c^2$, then $c=$

If the point of contact of the circles $x^2+y^2-6 x-4 y+9=0$ and $x^2+y^2+2 x+2 y-7=0$ is $(\alpha, \beta)$, then $7 \beta=$

If the circles $x^2+y^2-2 \lambda x-2 y-7=0$ and $3\left(x^2+y^2\right)-8 x+29 y=0$ are orthogonal, then $\lambda=$

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

If the circle passing through $(3,5),(5,5)$ and $(3,-3)$ cuts the circle $x^2+y^2+2 x+2 f y=0$ orthogonally, then $f=$

Length of the common chord of two circles of same radius is $2 \sqrt{17}$. If one of the two circles is $x^2+y^2+6 x+4 y-12=0$, then acute angle between the two circles is

A circle $S \equiv x^2+y^2-16=0$ intersects another circle $S^{\prime}=0$ of radius 5 units such that their common chord is of maximum length. If the slope of that chord is $\frac{3}{4}$, then the centre of such a circle $S^{\prime}=0$ is

Let $\theta$ be the angle between the circles $S \equiv x^2+y^2+2 x-2 y+c=0$ and $S^{\prime} \equiv x^2+y^2-6 x-8 y+9=0$. If $c$ is an integer and $\cos \theta=\frac{5}{16}$, then the radius of the circle $S=0$ is

If a circle $S$ passes through the origin and makes an intercept of length 4 units on the line $x=2$, then the equation of the curve on which the centre of $S$ lies is

A circle touches the line $2 x+y-10=0$ at $(3,4)$ and passes through the point $(1,-2)$. Then, a point that lies on the circle is

If $(a, b)$ is the common point for the circles $x^2+y^2-4 x+4 y-1=0$ and $x^2+y^2+2 x-4 y+1=0$, then $a^2+b^2=$

The angle between the tangents drawn from the point $(2,2)$ to the circle $x^2+y^2+4 x+4 y+c=0$ is $\cos ^{-1}\left(\frac{7}{16}\right)$. If two such circles exist, then sum of the values of $c$ is

If the circle $S=x^2+y^2+2 g x+4 y+1=0$ bisects the circumference of the circle $x^2+y^2-2 x-3=0$, then the radius of circle $S=0$ is

From a point $P$ on the circle $x^2+y^2=4$, two tangents are drawn to the circle $x^2+y^2-6 x-6 y+14=0$. If $A$ and $B$ are the points of contact of those lines, then the locus of the centre of the circle passing through the points $P$, $A$ and $B$ is

If the product of the lengths of the perpendicular drawn from the ends of a diameter of the circle $x^2+y^2=4$ on the line $x+y+1=0$ is maximum, then the two ends of that diameter are

If the intercept made by a variable circle on the X -axis and $Y$-axis are 8 and 6 units respectively, then the locus of the centre of the circle is

The slope of the non-vertical tangent drawn from the point $(3,4)$ to the circle $x^2+y^2=9$ is

If the acute angle between the circles $S \equiv x^2+y^2+2 k x+4 y-3=0$ and $S^{\prime} \equiv x^2+y^2-4 x+2 k y+9=0$ is $\cos ^{-1}\left(\frac{3}{8}\right)$ and the centre of $S^{\prime}=0$ lies in the first quadrant, then the radical axis of $S=0$ and $S^{\prime}=0$ is

If the line through the point $P(5,3)$ meets the circle $x^2+y^2-2 x-4 y+\alpha=0$ at $A(4,2)$ and $B\left(x_1, y_1\right)$, then $P A \cdot P B$ is equal to

$C_1$ is the circle with centre at $O(0,0)$ and radius $4, C_2$ is a variable circle with centre at $(\alpha, \beta)$ and radius 5 . If the common chord of $C_1$ and $C_2$ has slope $\frac{3}{4}$ and of maximum length, then one of the possible values of $\alpha+\beta$ is

If the pair of tangents drawn to the circle $x^2+y^2=a^2$ from the point $(10,4)$ are perpendicular. then $a=$

If $x-4=0$ is the radical axis of two orthogonal cirlces out of which one is $x^2+y^2=36$, then the centre of the other circle is

The radius of the circle which cuts the circles $x^2+y^2-4 x-4 y+7=0, x^2+y^2+4 x-4 y+6=0$ and $x^2+y^2+4 x+4 y+5=0$ orthogonally is

Equation of the circle having its centre on the line $2 x+y+3=0$ and having the lines $3 x+4 y-18=0,3 x+4 y+2=0$ as tangents is

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

The radius of the circle having. $3 x-4 y+4=0$ and $6 x-8 y-7=0$ as its tangents 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 least distance of the point $(10,7)$ from the circle $x^2+y^2-4 x-2 y-20=0$ 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 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

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

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

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

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

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

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

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 points where the circle $x^2+y^2-3 x -4 y+2=0$ cuts the $X$-axis are

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

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

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 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

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 locus of a point, which is at a distance of 4 units from $(3,-2)$ in $x y$-plane 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 angle between the pair of tangents drawn from $(1,1)$ to the circle $x^2+y^2+4 x+4 y-1=0$ 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

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

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.

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 $x^2+y^2-2 x-15=0$

Given, two fixed points $A(-2,1)$ and $B(3,0)$. Find the locus of a point $P$ which moves such that the angle $\angle A P B$ is always a right angle.

The equations of the tangents to the circle $x^2+y^2=4$ drawn from the point $(4,0)$ are

If $P(-9,-1)$ is a point on the circle $x^2+y^2+4 x+8 y-38=0$, then find equation of the tangent drawn at the other end of the diameter drawn through $P$

Find the equation of a circle whose radius is 5 units and passes through two points on the $X$-axis, which are at a distance of 4 units from the origin

If a foot of the normal from the point $(4,3)$ to a circle is $(2,1)$ and $2 x-y-2=0$, is a diameter of the circle, then the equation of circle is

The length of the tangent from any point on the circle $(x-3)^2+(y+2)^2=5 r^2$ to the circle $(x-3)^2+(y+2)^2=r^2$ is 16 units, then the area between the two circles in square units is

The equation of the circle, which cuts orthogonally each of the three circles

$\begin{aligned} & x^2+y^2-2 x+3 y-7=0, \\ & x^2+y^2+5 x-5 y+9=0 \text { and } \\ & x^2+y^2+7 x-9 y+29=0 \end{aligned}$

Find the equations of the tangents drawn to the circle $x^2+y^2=50$ at the points where the line $x+7=0$ meets it.

If the chord of contact of tangents from a point on the circle $x^2+y^2=r_1^2$ to the circle $x^2+y^2=r_2^2$ touches the circle $x^2+y^2=r_3^2$, then $r_1, r_2$ and $r_3$ are in

Find the equation of the circle passing through $(1,-2)$ and touching the $X$-axis at $(3,0)$.

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 $L_1$

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.

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