Example 2 Given a circle centered at the origin with radius $a$. Let the intersection points of the circle with the $x$-axis be $A, A^{\prime}$, and the moving chord perpendicular to the diameter $A A^{\prime}$ be $Q Q^{\prime}$. Let the intersection point of $Q Q^{\prime}$ with the $x$-axis be $R$, and the midpoints of $R Q, R Q^{\prime}$ be $M, M^{\prime}$, respectively. Try to find the locus of the intersection point $P$ of the two lines $A^{\prime} M$ and $A M^{\prime}$.
As shown in Figure 11-2, let the coordinates of $A, A^{\prime}$ be $(a, 0), (-a, 0)$, and the coordinates of $Q, Q^{\prime}$ be $Q(a \cos \theta, a \sin \theta), Q^{\prime}(a \cos \theta, -a \sin \theta)$, respectively. Then the coordinates of $M, M^{\prime}$ are $M\left(a \cos \theta, \frac{1}{2} a \sin \theta\right)$ and $M^{\prime}\left(a \cos \theta, -\frac{1}{2} a \sin \theta\right)$. The equations of the lines $A M^{\prime}$ and $A^{\prime} M$ are:
$$
A M^{\prime}: y=\frac{-\frac{1}{2} a \sin \theta}{a \cos \theta - a}(x - a);
$$
$$
A^{\prime} M: y=\frac{\frac{1}{2} a \sin \theta}{a \cos \theta + a}(x + a).
$$
(1) $\times$ (2), we get
$$
y^{2}=\frac{-\frac{1}{4} a^{2} \sin ^{2} \theta}{a^{2}\left(\cos ^{2} \theta - 1\right)}\left(x^{2} - a^{2}\right)=\frac{1}{4}\left(x^{2} - a^{2}\right),
$$

Therefore,
$$
\frac{x^{2}}{a^{2}} - \frac{y^{2}}{\left(\frac{a}{2}\right)^{2}} = 1
$$

Thus, the required locus is a hyperbola (3).