Tracing the Conic

The foci of the locus which is an ellipse are at $(3,1)$ and $(1,3)$ . So it's center is at $(2,2)$ . If the semi-major axis be $a$ , then $2a=3\implies a=\dfrac{3}{2}$ . So semi-minor axis is given by $b=\sqrt {\left (\dfrac{3}{2}\right) ^2-( 2-3)^2-(2-1)^2}=\dfrac{1}{2}\implies e=\sqrt {1-\left (\dfrac{b}{a}\right) ^2}=\boxed {\dfrac{2\sqrt 2}{3}}$ where $e$ is the required eccentricity.