The image of the point $(3, 4)$ 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

In the figure shown,the radius of circle $C_1$ is $r$ and that of $C_2$ is $\frac{r}{2}$,where $r = \frac{1}{3} PQ$. Then the length of $AB$ is (where $P$ and $Q$ are the centers of $C_1$ and $C_2$ respectively).

Find the equation of the chord of the circle $x^2 + y^2 = 16$ which is bisected at the point $(2, -1)$.

Three circles of radii $a, b, c$ $(a < b < c)$ touch each other externally. If they have the $x$-axis as a common tangent,then:

The length of the chord intercepted by the circle $x^2+y^2-6x+8y-5=0$ on the line $2x-y=5$ is equal to $L$ units. Find $L$.

Let $S$ be a subset of the plane defined by $S = \{(x, y) : |x| + 2|y| = 1\}$. Then,the radius of the smallest circle with centre at the origin and having non-empty intersection with $S$ is