JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Circle

Here $S_1$ has centre $(0,0)$ and radius $3$, and $S_2$ has centre $(2,0)$ and radius $1$. If the variable circle has centre $C$ and radius $r$, internal contact with $S_1$ gives $\left|C-(0,0)\right|=3-r$ and external contact with $S_2$ gives $\left|C-(2,0)\right|=1+r$. Adding these removes $r$:

So $C$ lies on an ellipse with foci $(0,0)$ and $(2,0)$, centre $(1,0)$, $a=2$, $c=1$ and $b^2=a^2-c^2=3$:

Putting $x=2$ gives $\frac{1}{4}+\frac{y^2}{3}=1$, so $y=\pm \frac{3}{2}$. The locus always passes through $\left(2,\pm \frac{3}{2}\right)$.