If the lines $3x - 4y + 4 = 0$ and $6x - 8y - 7 = 0$ are tangents to a circle, then the radius of the circle is

A circle with centre $(2,3)$ and radius $4$ intersects the line $x + y =3$ at the points $P$ and $Q$. If the tangents at $P$ and $Q$ intersect at the point $S(\alpha, \beta)$, then $4 \alpha-7 \beta$ is equal to $........$.

Which of the following lines is a tangent to the circle ${x^2} + {y^2} = 25$ for all values of $m$.....

Tangents are drawn from the point $(-1,-4)$ to the circle $x^2 + y^2 - 2x + 4y + 1 = 0$. Length of corresponding chord of contact will be-

Let the normals at all the points on a given curve pass through a fixed point $(a, b) .$ If the curve passes through $(3,-3)$ and $(4,-2 \sqrt{2}),$ and given that $a-2 \sqrt{2} b=3,$ then $\left(a^{2}+b^{2}+a b\right)$ is equal to ..... .

The area of the triangle formed by the tangents from the points $(h, k)$ to the circle ${x^2} + {y^2} = {a^2}$ and the line joining their points of contact is