Question

20 POINTS ASAP ALGEBRA II

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 $x = \frac{25}{4}.$ The point $P$ traces a curve in the plane, so that
\[\frac{PF}{PQ} = \frac{4}{5}\]for all points $P$ on the curve. Find the area of the region formed by the curve.

Answer 1

The curve described described here is an ellipse with eccentricity $e=\dfrac45$, the ratio between the distance from the given focus F(4, 0) to any point Q along the directrix $x=\dfrac{25}4$. Use this info to find the lengths of the semimajor and -minor axes, $a$ and $b$. Then the area of the ellipse is $\pi ab$.