Number of integral points interior to the circle $x^2 + y^2 = 10$ from which exactly one real tangent can be drawn to the curve $\sqrt {{{\left( {x + 5\sqrt 2 } \right)}^2} + {y^2}} \, - \sqrt {{{\left( {x - 5\sqrt 2 } \right)}^2} + {y^2}\,} \, = 10$ are (where integral point $(x, y)$ means $x, y  \in  I)$

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

If the line $lx + my = 1$ be a tangent to the circle ${x^2} + {y^2} = {a^2}$, then the locus of the point $(l, m)$ is

Let $PQ$ and $RS$ be tangents at the extremeties of the diameter $PR$ of a circle of radius $r$. If $PS$ and $RQ$ intersect at a point $X$ on the circumference of the circle, then $2r$ equals

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 area of the triangle formed by the positive $x$-axis and the normal and the tangent to the circle $x^2 + y^2 = 4$ at $(1, \sqrt 3 )$ is