Lines are drawn from a point $P (-1, 3)$ to a circle $x^2 + y^2 - 2x + 4y - 8 = 0$. Which meets the circle at $2$ points $A$ & $B$, then the minimum value of $PA + PB$ is

The equations of the tangents to the circle ${x^2} + {y^2} = 36$ which are inclined at an angle of ${45^o}$ to the $x$-axis are

The line $2 x - y +1=0$ is a tangent to the circle at the point $(2,5)$ and the centre of the circle lies on $x-2 y=4$. Then, the radius of the circle is

Let $O$ be the origin and $OP$ and $OQ$ be the tangents to the circle $x^2+y^2-6 x+4 y+8=0$ at the point $P$ and $Q$ on it. If the circumcircle of the triangle OPQ passes through the point $\left(\alpha, \frac{1}{2}\right)$, then a value of $\alpha$ 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

$y - x + 3 = 0$ is the equation of normal at $\left( {3 + \frac{3}{{\sqrt 2 }},\frac{3}{{\sqrt 2 }}} \right)$ to which of the following circles