If $\sin ^{-1}(a)$ is the acute angle between the curves $x^2+y^2=4x$ and $x^2+y^2=8$ at the point $(2,2)$,then $a$ is equal to

If $\theta$ is the angle between the tangents from $(-1, 0)$ to the circle $x^2+y^2-5x+4y-2=0$,then $\theta$ is equal to

If the lines $12x - 5y - 17 = 0$ and $24x - 10y + 44 = 0$ are tangents to the same circle,then the radius of the circle is:

$S_1$ and $S_2$ are two concentric circles of radii $1$ and $2$ respectively. Two parallel tangents to $S_1$ cut off an arc from $S_2$. The length of the arc is

Let $6$ and $8$ be the $X$ and $Y$-intercepts made by the circle $S \equiv x^2+y^2+2gx+2fy+c=0$ respectively. If $gx+fy+1=0$ is a line passing through the point $(1, -1)$,then the radius of the circle $S=0$ is

The straight line $(x - 2) + (y + 3) = 0$ cuts the circle $(x - 2)^2 + (y - 3)^2 = 11$ at