10. In the Cartesian coordinate system, there are two fixed points $A, B$ 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 of line $l$ and line $B C$.
10. Let the coordinates of $A, B, C$ be $(a, 0), (b, 0), (0, c)$, 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 $a\left(x^{2}-b x\right)+b y^{2}=0$, which can be rewritten as $\frac{\left(x-\frac{b}{2}\right)^{2}}{\frac{b^{2}}{4}}+\frac{y^{2}}{\frac{a b}{4}}=1$ (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 $BC$ as $c$ tends to infinity. When $a b>0$, the trajectory is an ellipse. When $a b<0$, the trajectory is a hyperbola.