Interaction between Two Circles

Two circles touch each other internally. The radii of these circles are $2 \mathrm{cm}$  and $3 \mathrm{cm}$ respectively. Find the length of the largest chord of the outer circle which touches the internal circle at a point?$$

The radius of the circle, having centre at (2, 1), whose one of the chord is a diameter of the circle ${\text{x}}^2+{\text{y}}^2-2\text{x}-6\text{y}+6=0$

If the circles $x^2+y^2=9$ and $x^2+y^2+2\alpha x+2y+1=0$ touch each other internally, then $\alpha$ is equal to

If the radical centre of the following circles: $x^2+y^2+4x-7=0, 2x^2+2y^2+3x+5y-9=0, x^2+y^2+y=0$ is $\left(a,b\right)$ then find the value of $a+b$.

The equation of the radical axis of the circles: $x^2+y^2+2x+4y+1=0$ and $x^2+y^2+4x+y=0$ is

The equation of the radical axis of the circles: $x^2+y^2-3x-4y+5=0,$ and $3\left(x^2+y^2\right)-7x+8y-11=0$ is

Examine whether the circles $x^2+y^2=4$ and $x^2+y^2-2y-4x=20$ touch each other.

Suppose that two circles $C_1$ and $C_2$ in a plane have no points in common, then

Find the equation of the circle passing through $(1, 1)$ and the points of intersection of the circles $x^2+y^2+13x-3y=0$ and $2x^2+2y^2+4x-7y-25=0$.

Find the equation of the circle passing through the points of intersection of the circles $x^2+y^2-x+7y-3=0$ and $x^2+y^2-5x-y+1=0$ and with its centre on the line $y+x=0$.

Prove that the two circles $x^2+y^2-4=0$ and $2x^2+2y^2-11=0$ are concentric.

Show that the circles ${\mathrm{x}}^2+{\mathrm{y}}^2-8\mathrm{x}-4\mathrm{y}-16=0$ and $x^2+y^2-2x+4y+4=0$ touch internally and find the co-ordinates of their point of contact.

Three circles of centres $C_1,C_2,C_3$ and radii $3,4,5$ respectively touches each other externally. If $P$ is the point of intersection of tangents at the point of contact of circles, then the value of ${\left|P{C}_1\right|}^2+{\left|P{C}_2\right|}^2+{\left|P{C}_3\right|}^2$ is

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

The radical axis of any two circles is ________ to the line joining their centres

From a point $\left(h,0\right)$ common tangents are drawn to circles $x^2+y^2=1$ and ${\left(x-2\right)}^2+y^2=4$, value of $h$ is

If $S, S_1, S_2$ be the circles of radii $5,3,2$ respectively. If $S_1$ and $S_2$ touch externally and they touch internally with $S.$ The radius of circle $S_3$ which touches externally with $S_1$ and $S_2$ and internally with $S$ is

Three circle touch each other externally. A triangle is formed when the centre of these circles are joined together. Find the radii of the circle, if side of the triangle formed are $6 \mathrm{cm}, 8 \mathrm{cm}$ and $9 \mathrm{cm}$.

Choose the appropriate option for the given two circles $x^2 + y^2 + 2ax + b = 0$ and $x^2 + y^2 + 2cx + b = 0$, if they touch each other :

Given that from the point $(h, k),$ the tangents drawn to the circles
${\omega }_1:x^2+y^2=1$
${\omega }_2:x^2+y^2+8x+15=0$
${\omega }_3:x^2+y^2+10y+24=0$
are of equal length. Evaluate $hk.$