If an ellipse sides between two perpendicular lines, then the locus of the centre is

Let $2a$, $2b$ be the length of the major and minor axes respectively of the ellipse. If the ellipse slides between two perpendicular lines, the point of intersection $P$ of these lines being the point of intersection of perpendicular tangents lies on the Director circle of the ellipse.

This means that the centre $C$ of the ellipse is always at a constant distance $\sqrt{a^2+b^2}$ from $P$. Hence the locus of $C$ is a circle.