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 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

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

The angle between circles $x^2+y^2+2x+4y+1=0$ and $x^2+y^2-2x+6y-3=0$ is:

If $S=x^2+y^2+2x+17y+4=0, S^{\text{'}}=x^2+y^2+7x+6y+11=0$ and $S=x^2+y^2+2x+17y+4=0, S^{\text{'}}=x^2+y^2+7x+6y+11=0$ $y^2-x+22y+3=0$ are three circles, then the length of tangent from their radical center to $S=0$ units

The perpendicular distance from the point $\left(1,2\right)$ to common chord of the circles $2 x+4 y-4=0$ and $x^2+y^2+4x-6y-3=0$ is units.

Find the equation of a circle which cuts the circle $x^2+y^2-6\mathrm{x}+4\mathrm{y}-3=0$ orthogonally, while passing through $(3, 0)$ and touching the $y$-axis.

Choose the correct option regarding the following statements:
Statement- $1:$ The length of common chord of the circles $x^2+y^2+ax+by+c=0$ and
$x^2+y^2+bx+ay+c=0 \mathrm{equals} \frac{\sqrt{(a+b{)}^2-8\mathrm{c}}}{2}$
Statement-$2 :$ If two circles intersect at two distinct points, then their radical axis is their common chord.

If the circles $x^2+y^2-6x-8y-12=0$ and $x^2+y^2-4x+6y+k=0$ are perpendicular to each other, then $\text{'}k\text{'}$' equals

Which circle among the following bisects the circumference of the circle $x^2+y^2-8x-6y+23=0 ?$

If $L_1$ represents the radical axis of circles $x^2+y^2-4x-6y+5=0$ and $x^2+y^2-2x-4y-1=0$  and $L_2$ represents the radical axis of $x^2+y^2+2x+2y-7=0$ and $x^2+y^2+x+y+9=0,$ then $\_\_\_\_\_\_\_$

Equation of the straight line meeting the circle with centre at origin and radius equal to $5$ in two points at equal distances of $3$ units from the point $A\left(3, 4\right)$ is

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 common chord of the pair of circles: $(x-a{)}^2+(y-b{)}^2=c^2,$ and $(x-b{)}^2+(y-a{)}^2=c^2(a\ne b)$ is

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