13大 $\star \star$ Two flagpoles of heights $h$ and $k$ are erected on a horizontal plane, separated by $2a$ units. Find: the set of all points on the plane from which the angles of elevation to the tops of the poles are equal.
In a rectangular coordinate system, let the base of a flagpole of height $h$ be at $(-a, 0)$, and the base of a flagpole of height $k$ be at $(a, 0)$. Suppose point $p(x, y)$ has equal angles of elevation to the tops of both flagpoles, then $\frac{h}{\sqrt{(x+a)^{2}+y^{2}}} = \frac{k}{\sqrt{(x-a)^{2}+y^{2}}}$, which simplifies to $\left(k^{2}-h^{2}\right) x^{2}+2\left(k^{2}+h^{2}\right) a x+\left(k^{2}-h^{2}\right) y^{2}+a^{2}\left(k^{2}-h^{2}\right) = 0$.
(1) If $k=h$, the equation simplifies to $x=0$. Thus, the required locus is the $y$-axis.
(2) If $k \neq h$, the equation simplifies to $x^{2}+2 a x\left(\frac{k^{2}+h^{2}}{k^{2}-h^{2}}\right)+y^{2}+a^{2}=0$, and its locus is a circle.