5. Given points $A(0,3), B(0,6), C(0,-2), D(0,2)$, point $P$ is a moving point, and satisfies $\frac{|P A|}{|P B|}=\frac{1}{2}$. Let line $l$ be the perpendicular bisector of $P C$, and it intersects line $P D$ at point $Q$. Then, as point $P$ moves, the equation of the trajectory of point $Q$ is $\qquad$
As shown in the figure, the trajectory of point $P$ is a circle with $D$ as the center and a radius of 2. Clearly, $|Q C|=|Q P|$. When $D$ is on the line segment $Q P$, $|Q C|-|Q D| = |P D| = 2$; when $D$ is on the extension of the line segment $Q P$, $|Q D|-|Q C| = |P D| = 2$.
Thus, for the hyperbola, $2a = 2, c = 2 \Rightarrow a = 1, b = \sqrt{3}$, so the equation of the trajectory of point $Q$ is $y^{2}-\frac{x^{2}}{3}=1$.