The length of the tangent from the point $(5, 1)$ to the circle $x^2 + y^2 + 6x - 4y - 3 = 0$ is:

If $Q(h, k)$ is the inverse point of the point $P(1, 2)$ with respect to the circle $x^2+y^2-4x+1=0$,then $2h+k=$

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).

$A$ straight line meets the coordinate axes at $A$ and $B$. $A$ circle is circumscribed about the triangle $OAB$, where $O$ is the origin. If $m$ and $n$ are the distances of the tangent to the circle at the origin from the points $A$ and $B$ respectively, then the diameter of the circle is

The number of common tangents of the circles $x^2+y^2-4=0$ and $x^2+y^2-6x-8y-24=0$ is

$A$ line is drawn through a fixed point $P(\alpha, \beta)$ to cut the circle $x^{2}+y^{2}=r^{2}$ at $A$ and $B$. Then $PA \cdot PB$ is equal to