Let S _ 1 : x ^ 2 + y ^ 2 = 9 and S _ 2 : ( x - 2 ) ^ 2 + y ^ 2 = 1...

$∵ {c}_{1}{c}_{2} = {r}_{1} - {r}_{2}$

$∴$ Given circles are touching internally

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

$⇒ PA = {r}_{1} - r and PB = {r}_{2} + r$

$⇒ PA + PB = {r}_{1} + {r}_{2}$

$⇒ PA + PB = 4 ( > AB )$

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

$⇒$ 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

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

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