= Solution
A ray leaving focus $A$ and reaching a boundary point $P$ has incoming unit direction $\mathbf u_A$. By the <law of reflection>, its normal component reverses and its tangential component stays fixed. The result just derived shows that the reflected direction is $-\mathbf u_B$, pointing exactly from $P$ to focus $B$. Thus all such in-plane rays from one focus converge to the other after one reflection, in the ideal <geometrical optics> model.
Moreover, every reflected path has the same total length $AP+PB=s$. In a uniform medium their propagation times are equal as well, so different reflection points do not introduce a geometrical path-length spread. Finite <wavelength>, surface errors and incomplete reflecting coverage limit a real focus; the ideal argument concerns rays in the plane of the <ellipse>. It does not assert that an arbitrary three-dimensional surface with only one elliptical section focuses every out-of-plane ray.
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