Example 3 Let the equation of the ellipse be $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$, the major axis be $A A^{\prime}, P$ be any point on the ellipse, draw $A Q \perp A P$, $A^{\prime} Q \perp A P, A Q$ and $A^{\prime} Q$ intersect at $Q$, find the locus of $Q$.