The straight line $x+2 y=1$ cuts the $X$-axis at $A$ and $Y$-axis at $B, A$ circle is drawn through $A, B$ and the origin. The sum of the perpendicular distances from $A$ and $B$ on to the tangent drawn at origin to the circle $S$ is

Let $P$ and $Q$ be two external points of the circle $S=x^2+y^2-a^2=0$. Let the chord of contact of the point $P$ with respect to the circle $S=0$ passes through $Q$. If $l_1$ and $l_2$ are the lengths of the tangents drawn from $P$ and $Q$ to the circle $S=0$, then $P Q=$

$A\left(x_1, y_1\right)$ is the internal centre of similitude and $B\left(x_2, y_2\right)$ is the external centre of similitude of two circles $C_1$ and $C_2$ whose centes are $P(\alpha, \beta)$ and $Q(\gamma, \delta)$, respectively. If $P A=3, A B=5, Q B=2$, then ratio of the radii of the two circles is

The equation of the direct common tangent of the circles $x^2+y^2-6 x-4 y-23=0$ and $x^2+y^2+2 x+2 y+1=0$ is