13.3.6 ** Given $P(3,4)$ is a point inside the circle $x^{2}+y^{2}=64$, and two moving points $A$ and $B$ on the circumference of the circle satisfy $\angle A P B-\frac{\pi}{2}$. A rectangle $A P B Q$ is constructed with $A P$ and $B P$ as adjacent sides. Find the equation of the trajectory of point $Q$.
In $\triangle O P Q$, by applying the median length theorem, we know $|O P|^{2} +|O Q|^{2}=2\left(|O R|^{2}+|P R|^{2}\right)$, where $R$ is the intersection point of the diagonals of rectangle $A P B Q$. Since the diagonals of rectangle $A P B Q$ are equal in length, i.e., $|A B|=|P Q|$, then $|P R|=|A R|$, thus $|O P|^{2}+|O Q|^{2}-2\left(|O R|^{2}+|A R|^{2}\right)$. Also, since $R$ is the midpoint of chord $A B$, then $|O R|^{2}+|A R|^{2}=|O A|^{2}$, hence $|O P|^{2}+|O Q|^{2}=2|O A|^{2}$.

Let the coordinates of point $Q$ be $(x, y)$. Therefore, $3^{2}+4^{2}+x^{2}+y^{2}=$ $2 \times 64=128$, i.e., $x^{2}+y^{2}=103$.