jmc

Question

In the coordinate plane, let F = (5,0). Let P be a point, and let Q be the projection of the point P onto the line x = 165. The point P traces a curve in the plane, so that PFPQ = 54for all points P on the curve. Find the equation of this curve. (Enter it in standard form.) FIGURE

Accepted Answer

Let P = (x,y). Then Q = ( 165, y ), so the equation PFPQ = 54 becomes (x - 5)^2 + y^2| x - 165 | = 54.Then (x - 5)^2 + y^2 = | 54 x - 4 |, so 4 (x - 5)^2 + y^2 = |5x - 16|.Squaring both sides, we get 16x^2 - 160x + 16y^2 + 400 = 25x^2 - 160x + 256.This simplifies to 9x^2 - 16y^2 = 144,so x^216 - y^29 = 1.Thus, the curve is a hyperbola.