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MATH / 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=(25)/(4). The point P traces a curve in the plane, so that …

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Answer published by the source. Consult the official source to check your work against its answer.

problem

In the coordinate plane, let F=(4,0).F = (4,0). Let PP be a point, and let QQ be the projection of the point PP onto the line x=254.x = \frac{25}{4}. The point PP traces a curve in the plane, so that PFPQ=45\frac{PF}{PQ} = \frac{4}{5}for all points PP on the curve. Find the area of the region formed by the curve. [asy] unitsize(1 cm); pair P, F, Q; F = (4,0); P = (5*Cos(60),3*Sin(60)); Q = (25/4,3*Sin(60)); draw(F--P--Q); draw((25/4,-1)--(25/4,3),dashed); dot("FF", F, S); dot("PP", P, W); dot("QQ", Q, E); label("x=254x = \frac{25}{4}", (25/4,-1), S); [/asy]
Plain-text mathematical notation (without MathML)
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=(25)/(4).  The point P traces a curve in the plane, so that
(PF)/(PQ)=(4)/(5)for all points P on the curve.  Find the area of the region formed by the curve.

[asy]
unitsize(1 cm);

pair P, F, Q;

F = (4,0);
P = (5*Cos(60),3*Sin(60));
Q = (25/4,3*Sin(60));

draw(F--P--Q);
draw((25/4,-1)--(25/4,3),dashed);

dot("F", F, S);
dot("P", P, W);
dot("Q", Q, E);

label("x=(25)/(4)", (25/4,-1), S);
[/asy]
Original LaTeX notation
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.

[asy]
unitsize(1 cm);

pair P, F, Q;

F = (4,0);
P = (5*Cos(60),3*Sin(60));
Q = (25/4,3*Sin(60));

draw(F--P--Q);
draw((25/4,-1)--(25/4,3),dashed);

dot("$F$", F, S);
dot("$P$", P, W);
dot("$Q$", Q, E);

label("$x = \frac{25}{4}$", (25/4,-1), S);
[/asy]

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
939efc4a-77dd-5744-9152-ea0a53018fe8
task_key
intermediate~5falgebra--test--939efc4a-77dd-5744-9152-ea0a53018fe8
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import