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 centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance of $6$ units from each other. Let $P$ be the midpoint of the line segment joining the centres of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_1$ and $C$ passing through $P$ is also a common tangent to $C_2$ and $C$, then the radius (in units) of the circle $C$ is 

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

The number of common tangents to the circles $x^2+y^2+2x+8y-23=0$ and $x^2+y^2-4x-10y+9=0$, are

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

Find the number of common tangent(s) to the circles $x^2+y^2+2x+8y-23=0$ and $x^2+y^2-4x-10y+19=0$.

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

Area of triangle formed by common tangents to the circle $x^2+y^2-6x=0$ and $x^2+y^2+2x=0$ is -

If tangents be drawn to the circle $x^2+y^2=12$ at its points of intersection with the circle $x^2+y^2-$ $5 x+3 y-2=0,$ then the tangents intersect at the point