Let $P(4 \cos \theta, 3 \sin \theta), Q(x, y)$, then the equations of $P A$ and $P A^{\prime}$ are $y=\frac{3 \sin \theta}{4 \cos \theta-4}(x-4)$ and $y=\frac{3 \sin \theta}{4 \cos \theta+4}(x+4)$. From $Q A \perp P A, Q A^{\prime} \perp P A$ we get the equations of $Q A$ and $Q A^{\prime}$ as $y=\frac{4-4 \cos \theta}{3 \sin \theta}(x-4)$ and $y=\frac{4 \cos \theta+4}{3 \sin \theta}(x+4)$.

Multiplying the two equations and using $\sin ^{2} \theta=1-\cos ^{2} \theta$ we get $y^{2}=-\frac{16}{9}\left(x^{2}-16\right)$, rearranging gives $\frac{x^{2}}{16}+\frac{y^{2}}{\frac{256}{9}}=1$, which is the equation of the locus of the moving point $Q$.
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 $Q$.