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

If the common chord of the circles $x^2+y^2+4y=0$ and $x^2+y^2-4x-5=0$ is the diameter of the circle $S=0$ then the abscissa of the centre of the circle $S=0$ is

If the circles ${\mathrm{C}}_1:x^2+y^2+2x+4y-20=0, {\mathrm{C}}_2:x^2+y^2+6x-8y+9=0$ have $n$ common tangents and the length of the tangent drawn from the centre of similitude to the circle $C_2$ is $l$ then $\frac{l}{n^2}=$

If $\left(0,\frac{3}{4}\right)$ is the radical centre of the circles $S\equiv x^2+y^2+\alpha x+6y=0,S^{\text{'}}\equiv x^2+y^2+2\alpha x+\alpha y+6=0$ and $S^{"}\equiv x^2+y^2+6\alpha x-\alpha y+3=0$ then the distance between the radical centre and the centre of the circle $S^{\text{'}}=0$ is

If $\theta$ is the angle between the circles $x^2+y^2-2x-4y-4=0$ and $x^2+y^2-8x-12y+43=0$ then $\left|7\sec \theta -18\cos \theta \right|=$

The equation of the transverse common tangent of the circles $x^2+y^2-6x-8y+9=0$ and $x^2+y^2+2x-2y+1=0$ is

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

The equation of the circle which cuts orthogonally the circle $x^2+y^2-4x+2y-7=0$ and having the centre at $(2,3)$ is

The equation of the circle which passes through the points $(2,0),(0,2)$ and orthogonal to the circle $2x^2+2y^2+5x-6y+4=0$ is

The equation of the circle passing through the origin, having its centre on the line $x+y=4$ and intersecting the circle $x^2+y^2-4x+2y+4=0$ orthogonally is

The equation of the circle which passes through the origin and intersects the circles, $x^2+y^2-4x-6y-3=0,$ and $x^2+y^2-8y+12=0$ orthogonally is