A circle $\mathrm{C}$ touches the line $\mathrm{x}=2 \mathrm{y}$ at the point $(2,1)$ and intersects the circle $C_{1}: x^{2}+y^{2}+2 y-5=0$ at two points $\mathrm{P}$ and $\mathrm{Q}$ such that $\mathrm{PQ}$ is a diameter of $\mathrm{C}_{1}$. Then the diameter of $\mathrm{C}$ is :

Give the number of common tangents to circle ${x^2} + {y^2} + 2x + 8y – 23 = 0$ and ${x^2} + {y^2} – 4x – 10y + 9 = 0$

If the variable line $3 x+4 y=\alpha$ lies between the two circles $(x-1)^{2}+(y-1)^{2}=1$ and $(x-9)^{2}+(y-1)^{2}=4$ without intercepting a chord on either circle, then the sum of all the integral values of $\alpha$ is …. .

The number of direct common tangents to the circles $x^2 + y^2 = 4$ and $x^2 + y^2 -8x -8y + 7 = 0$ , is

Let $C_1, C_2$ be two circles touching each other externally at the point $A$ and let $A B$ be the diameter of circle $C_1$. Draw a secant $B A_3$ to circle $C_2$, intersecting circle $C_1$ at a point $A_1(\neq A)$, and circle $C_2$ at points $A_2$ and $A_3$. If $B A_1=2, B A_2=3$ and $B A_3=4$, then the radii of circles $C_1$ and $C_2$ are respectively