Let $P$ be a point on the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ and the line through $P$ parallel to the $Y$-axis meets the circle $x^{2}+y^{2}=9$ at $Q$, where $P$ and $Q$ are on the same side of the $X$-axis. If $R$ is a point on $PQ$ such that $\frac{PR}{RQ}=\frac{1}{2}$, then the locus of $R$ is

$A$ point moves in such a way that its distance from the origin is always $4$. Then the locus of the point is:

Let $T'$ be the line passing through the points $P(-2, 7)$ and $Q(2, -5)$. Let $F_1$ be the set of all pairs of circles $(S_1, S_2)$ such that $T'$ is tangent to $S_1$ at $P$ and tangent to $S_2$ at $Q$,and also such that $S_1$ and $S_2$ touch each other at a point,say,$M$. Let $E_1$ be the set representing the locus of $M$ as the pair $(S_1, S_2)$ varies in $F_1$. Let the set of all straight line segments joining a pair of distinct points of $E_1$ and passing through the point $R(1, 1)$ be $F_2$. Let $E_2$ be the set of the mid-points of the line segments in the set $F_2$. Then,which of the following statement$(s)$ is (are) $TRUE$?

From the origin,chords are drawn to the circle $(x - 1)^2 + y^2 = 1$. The equation of the locus of the midpoints of these chords is

The angle between a pair of tangents drawn from a point $P$ to the circle ${x^2} + {y^2} + 4x - 6y + 9{\sin ^2}\alpha + 13{\cos ^2}\alpha = 0$ is $2\alpha$. The equation of the locus of the point $P$ is

Let $Q$ be a point on the circle $B: x^2+y^2=a^2$ and $P(h, k)$ be a fixed point. If the locus of the point which divides the join of $P$ and $Q$ in the ratio $p: q$ is a circle $C$,then the centre of $C$ is