Interaction between Two Circles

If the circle $C_1:x^2+y^2=16$ intersects another circle $C_2$ of radius $5$ in such a manner that the common chord is of maximum length and has a slope equal to $\frac{3}{4},$ then the coordinates of the centre of $C_2$ are

Prove that the two circles $x^2+y^2+2by+c^2=0\text{ and }{x}^2+y^2+2ax+c^2=0$ will touch each other if $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}$.

The equation to the circle cutting orthogonally the three circles $x^2+y^2-2x+3y-7=0, x^2+y^2+5x-5y+9=0$ and $x^2+y^2+7x-9y+29=0$, is

The number of common tangents to the circles $x^2+y^2-4x-6y-12=0$ and $x^2+y^2+6x+18y+26=0$ is

Let circles $S_1$ and $S_2$ of radii $r_1$ and $r_2$ respectively $\left(r_1>r_2\right)$ touches each other externally. Circle $S$ of radius $r$ touches $S_1$ and $S_2$ externally and also their direct common tangent. Prove that the triangle formed by joining centre of $S_1, S_2$ and $S$ is obtuse angled triangle.

Show that if one of the circle $x^2+y^2+2gx+c=0$ and $x^2+y^2+2g_{1}x+c=0$ lies within the other, then $g{g}_1$ and $C$ are both positive.

Find the equation of the circle which cuts each of the circles, $x^2+y^2=4, x^2+y^2-6x-8y+10=0$ $\& x^2+y^2+2x-4y-2=0$ at the extremities of a diameter.

$P\left(a, b\right)$ is a point in the first quadrant. If the two circles which pass through $P$ and touch both the co-ordinate axes cut at right angles, then find the condition in $a$ and $b$.

Consider points $A\left(\sqrt{13}, 0\right)$ and $B\left(2\sqrt{13}, 0\right)$ lying on $x$-axis. These points are rotated in an anticlockwise direction about the origin through an angle of ${\tan }^{-1}\left(\frac{2}{3}\right)·$ Let the new position of $A$ and $B$ be $A^{\text{'}}$ and $B^{\text{'}}$, respectively. With $A^{\text{'}}$ as centre and radius $\frac{2\sqrt{13}}{3}$ a circle $C_1$ is drawn and with $B^{\text{'}}$ as a centre and radius $\frac{\sqrt{13}}{3}$ circle $C_2$ is drawn. Find the radical axis of $C_1$ and $C_2$

The centre of the circle $S=0$ lies on the line $2x-2y+9=0$ and $S=0$ cuts orthogonally the circle $x^2+y^2=4.$ Show that the circle $S=0$ passes through two fixed points and also find their coordinates.

Circles are inscribed in the acute angle $\alpha$ so that every neighbouring circles touch each other. If the radius of the first circle is $R,$ then find the sum of the radii of the first $n$ circles in terms of $R$ and $\alpha .$

$x^2+y^2=a^2$ and $(x-2a{)}^2+y^2=a^2$ are two equal circles touching each other. Find the equation of circle (or circles) of the same radius touching both the circles.

Two circles, each of radius $5$ units, touch each other at $(1, 2).$ If the equation of their common tangent is $4x+3y=10.$ The equations of the circles are

Which of the following statement $\left(s\right)$ is/are correct with respect to the circles $S_1\equiv x^2+y^2-4=0$ and $S_2\equiv x^2+y^2-2x-4y+4=0?$

For the circles $x^2+y^2-10x+16y+89-r^2=0$ and $x^2+y^2+6x-14y+42=0$ which of the following is/are trüe?

The centre of a circle passing through the points $(0, 0), (1, 0) \&$ touching the circle $x^2+y^2=9$ is 

The circumference of the circle $x^2+y^2-2x+8y-q=0$ is bisected by the circle $x^2+y^2+4x+12y+p=0,$ then find $p+q$.

Two circles whose radii are equal to $4$ and $8$ intersect at right angles. The length of their common chord is $\frac{\lambda }{\sqrt{5}},$ then find $\lambda .$

The circle $x^2+y^2=4$ cuts the circle $x^2+y^2+2x+3y-5=0$ in $A \& B$. Then the equation of the circle on $AB$ as a diameter is

STATEMENT-$1$: If three circles which are such that their centres are non-collinear, then exactly one circle exists which cuts the three circles orthogonally.

STATEMENT-$2$: Radical axis for two intersecting circles is the common chord.