The locus of a point $P$ which divides the line joining $(1, 0)$ and $(2\cos \theta, 2\sin \theta)$ internally in the ratio $2 : 3$ for all $\theta$ is a

The number of solid cones with integer radius and integer height each having its volume numerically equal to its total surface area is

If a circle passes through the point $(1, 2)$ and intersects the circle $x^2 + y^2 = 4$ orthogonally,then the equation of the locus of its center is:

$A$ circle is drawn to cut a chord of length $2a$ units along the $X$-axis and to touch the $Y$-axis. The locus of the centre of the circle is

Three distinct points $A, B$,and $C$ are given in a two-dimensional coordinate plane such that for each point,the ratio of its distance from $(1, 0)$ to its distance from $(-1, 0)$ is equal to $\frac{1}{3}$. What is the circumcenter of triangle $ABC$?

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