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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.

User Joey Dalu
by
6.6k points

1 Answer

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The curve described described here is an ellipse with eccentricity
e=\frac45, the ratio between the distance from the given focus F(4, 0) to any point Q along the directrix
x=\frac{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.

User Lvarayut
by
6.8k points
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