Let $r_{1}$ and $r_{2}$ be the radii of the largest and smallest circles, respectively, which pass through the point $(-4,1)$ and having their centres on the circumference of the circle $x^{2}+y^{2}+2 x+4 y-4= 0.$ If $\frac{r_{1}}{r_{2}}=a+b \sqrt{2}$, then $a+b$ is equal to:

If two circles ${(x - 1)^2} + {(y - 3)^2} = {r^2}$ and ${x^2} + {y^2} - 8x + 2y + 8 = 0$ intersect in two distinct points, then

The radical axis of the pair of circle ${x^2} + {y^2} = 144$ and ${x^2} + {y^2} - 15x + 12y = 0$ is

Two circles whose radii are equal to $4$ and $8$ intersects at right angles. The length of  their common chord 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

If the circles ${x^2} + {y^2} + 2ax + cy + a = 0$ and ${x^2} + {y^2} - 3ax + dy - 1 = 0$ intersect in two distinct points $P$ and $Q$ then the line $5x + by - a = 0$ passes through $P$ and $Q$ for