# Tough Locus Problem

## Homework Statement

From a point P, perpendiculars PM and PN are drawn to two fixed straight lines OM and ON. If the area OMPN, be constant, prove that the locus of P is a hyperbola.

How do we start?

You can, without loss of generality, assume that one line, say OM, is the y-axis, and the other, ON, is given by y= mx for some number m. If P has coordinates $(x_0, y_0)$ then the perpendicular to OM is the line $y= y_0$. The line through P perpendicular to ON is given by $y= -(1/m)(x- x_0)+ y_0$. You can find its length by finding x such that $y= -(1/m)(x- x_0)+ y_0= mx$, where the two lines cross. Find the area of that figure, as a function of $x_0$ and $y_0$, set it equal to a constant, and see what relation you get between $x_0$ and $y_0$.