Let S 1   :   x 2 + y 2 = 9 and S 2   :   ( x - 2 ) 2 + y 2 = 1 . Then the locus of…

$\because c_1{c}_2=r_1-r_2$

$\therefore$ Given circles are touching internally

Let a variable circle with centre $P$ and radius $r$

$\Rightarrow PA=r_1-r$ and $PB=r_2+r$

$\Rightarrow PA+PB=r_1+r_2$

$\Rightarrow PA+PB=4(>AB)$

$\Rightarrow$ Locus of $P$ is an ellipse with foci at $A(0,0)$ and $B(2,0)$ and length of major axis is $2a=4$, $e=\frac{1}{2}$

$\Rightarrow$ Centre is at $(1,0)$ and $b^2=a^2\left(1-e^2\right)=3$

if $x$-axis is the axis of ellipse then

$\Rightarrow E : \frac{(x-1{)}^2}{4}+\frac{y^2}{3}=1$

which is satisfied by $\left(2,\pm \frac{3}{2}\right)$