12.28 ** In the Cartesian coordinate system, there are two fixed points $A$ and $B$ on the $x$-axis, different from the origin $O(0,0)$. A moving point $C$ varies on the $y$-axis, and a line $l$ passes through $O(0,0)$ and is perpendicular to the line $A C$. Find the locus of the intersection point of line $l$ and line $B C$.
Let the coordinates of $A$, $B$, and $C$ be $(a, 0)$, $(b, 0)$, and $(0, c)$, respectively. Given that $a \neq 0$, $b \neq 0$, $a \neq b$. When $c \neq 0$, the slope of $AC$ is $-\frac{c}{a}$, and the slope of $l$ is $\frac{a}{c}$, so the equation of $l$ is $y = \frac{a}{c} x$.
(1) The equation of $BC$ is $\frac{x}{b} + \frac{y}{c} = 1$.
(2) By eliminating $c$ from (1) and (2), we obtain the trajectory equation as $a(x^2 - bx) + by^2 = 0$, which can be rewritten as $\frac{(x - \frac{b}{2})^2}{\frac{b^2}{4}} + \frac{y^2}{\frac{ab}{4}} = 1$. (3) When $c = 0$, the intersection point of $l$ and $BC$ is the origin, which is also included in equation (3). The point $(b, 0)$ on equation (3) can be considered as the limit position of the intersection point of $l$ and $BC$ when $C$ tends to infinity. When $ab > 0$, the trajectory is an ellipse. When $ab < 0$, the trajectory is a hyperbola.