jmc

Question

In the coordinate plane, let F = (4,0). Let P be a point, and let Q be the projection of the point P onto the line x = 254. The point P traces a curve in the plane, so that PFPQ = 45for all points P on the curve. Find the area of the region formed by the curve. FIGURE

Accepted Answer

Let P = (x,y); then Q = ( 254, y ). The condition PFPQ = 45 becomes (x - 4)^2 +y^2|254 - x| = 45.Hence, (x - 4)^2 + y^2 = | 5 - 45 x |, or 5 (x - 4)^2 + y^2 = |25 - 4x|.Squaring both sides, we get 25 ((x - 4)^2 + y^2) = (25 - 4x)^2.This simplifies to 9x^2 + 25y^2 = 225, or x^225 + y^29 = 1.Hence, the curve is an ellipse, and its area is 5 3 = 15 .