Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S1 / 2 / d / Solution

For a radius $\rho >0$, write $z=\rho e^{i\theta }$. ThenFor $\rho \ne 1$ its image is the ellipseIts horizontal semiaxis is $\rho +{\rho }^{-1}$, its vertical semiaxis is $|\rho -{\rho }^{-1}|$, and its foci are $\pm 2$ because the squared semiaxes differ by $4$. Reciprocal radii give the same ellipse, traversed in opposite directions as $\theta$ increases.