In the given figure, $AB$ is tangent to the circle with centre $O$ , the ratio of the shaded region to the unshaded region of the triangle $OAB$ is

If the line $y = mx + c$be a tangent to the circle ${x^2} + {y^2} = {a^2}$, then the point of contact is

Let $B$ be the centre of the circle $x^{2}+y^{2}-2 x+4 y+1=0$ Let the tangents at two points $\mathrm{P}$ and $\mathrm{Q}$ on the circle intersect at the point $\mathrm{A}(3,1)$. Then $8.$ $\left(\frac{\text { area } \triangle \mathrm{APQ}}{\text { area } \triangle \mathrm{BPQ}}\right)$ is equal to .... .

The equation of three circles are ${x^2} + {y^2} - 12x - 16y + 64 = 0,$ $3{x^2} + 3{y^2} - 36x + 81 = 0$ and ${x^2} + {y^2} - 16x + 81 = 0.$ The co-ordinates of the point from which the length of tangent drawn to each of the three circle is equal is

If $2x - 4y = 9$ and $6x - 12y + 7 = 0$ are the tangents of same circle, then its radius will be

The equations of the tangents drawn from the origin to the circle ${x^2} + {y^2} - 2rx - 2hy + {h^2} = 0$ are