Tangents are drawn from the point $(4, 3)$ to the circle ${x^2} + {y^2} = 9$. The area of the triangle formed by them and the line joining their points of contact is

The angle between the two tangents from the origin to the circle ${(x - 7)^2} + {(y + 1)^2} = 25$ is

Suppose two perpendicular tangents can be drawn from the origin to the circle $x^2+y^2-6 x-2 p y+17=0$, for some real $p$. Then, $|p|$ is equal to

The line $y = x + c$will intersect the circle ${x^2} + {y^2} = 1$ in two coincident points, if

If the distances from the origin to the centres of the three circles ${x^2} + {y^2} - 2{\lambda _i}\,x = {c^2},(i = 1,\,2,\,3)$ are in $G. P.$, then the lengths of the tangents drawn to them from any point on the circle ${x^2} + {y^2} = {c^2}$ are in

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