$A$ piece of paper in the shape of a sector of a circle (see $Fig. 1$) is rolled up to form a right-circular cone (see $Fig. 2$). The value of the angle $\theta$ is

Let a point $P$ be such that its distance from the point $(5, 0)$ is thrice the distance of $P$ from the point $(-5, 0)$. If the locus of the point $P$ is a circle of radius $r$,then $4r^{2}$ is equal to ...... .

Let $S = 0$ be the locus of the center of a variable circle which intersects the circle $x^2 + y^2 - 4x - 6y = 0$ orthogonally at the point $(4, 6)$. If $P$ is a variable point on $S = 0$,then the least value of $OP$ is (where $O$ is the origin).

Tangents are drawn from the point $(17,7)$ to the circle $x^2+y^2=169$.
$STATEMENT-1$: The tangents are mutually perpendicular.
$STATEMENT-2$: The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is $x^2+y^2=338$.

The equation of the director circle of the circle ${x^2} + {y^2} = {a^2}$ is

$A$ circle touches the $X$-axis and also touches the circle with radius $2$ and center at $(0, 3)$. Find the locus of the center of the circle.