benchmarks.wiki

benchmarks.wiki / Public workspace

MATH / The foci of the ellipse (x²)/(49)+(y²)/(33)=1 are F₁ and F₂, as shown below. Let P be a point on the circle x²+(y−3)²=4. Line F₂P intersects the ellipse again at Q, where the y-coo…

Problem

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

problem

The foci of the ellipse x249+y233=1\frac{x^2}{49} + \frac{y^2}{33} = 1 are F1F_1 and F2,F_2, as shown below. Let PP be a point on the circle x2+(y−3)2=4.x^2 + (y - 3)^2 = 4. Line F2PF_2 P intersects the ellipse again at Q,Q, where the yy-coordinate of QQ is positive. Find the maximum value of PQ+F1Q.PQ + F_1 Q. [asy] unitsize(0.4 cm); pair P, Q; pair[] F; path ell = yscale(sqrt(33))*xscale(7)*Circle((0,0),1); F[1] = (4,0); F[2] = (-4,0); P = (0,3) + 2*dir(240); Q = intersectionpoint(P--interp(F[2],P,5),ell); draw(ell); draw((-8,0)--(8,0)); draw((0,-7)--(0,7)); draw(Circle((0,3),2)); draw(F[1]--Q--F[2]); dot("F1F_1", F[1], S); dot("F2F_2", F[2], S); dot("PP", P, S); label("QQ", Q, NE); [/asy]
Plain-text mathematical notation (without MathML)
The foci of the ellipse (x²)/(49)+(y²)/(33)=1 are F₁ and F₂, as shown below.  Let P be a point on the circle x²+(y−3)²=4.  Line F₂P intersects the ellipse again at Q, where the y-coordinate of Q is positive.  Find the maximum value of PQ+F₁Q.

[asy]
unitsize(0.4 cm);

pair P, Q;
pair[] F;

path ell = yscale(sqrt(33))*xscale(7)*Circle((0,0),1);
F[1] = (4,0);
F[2] = (-4,0);
P = (0,3) + 2*dir(240);
Q = intersectionpoint(P--interp(F[2],P,5),ell);

draw(ell);
draw((-8,0)--(8,0));
draw((0,-7)--(0,7));
draw(Circle((0,3),2));
draw(F[1]--Q--F[2]);

dot("F₁", F[1], S);
dot("F₂", F[2], S);
dot("P", P, S);
label("Q", Q, NE);
[/asy]
Original LaTeX notation
The foci of the ellipse $\frac{x^2}{49} + \frac{y^2}{33} = 1$ are $F_1$ and $F_2,$ as shown below.  Let $P$ be a point on the circle $x^2 + (y - 3)^2 = 4.$  Line $F_2 P$ intersects the ellipse again at $Q,$ where the $y$-coordinate of $Q$ is positive.  Find the maximum value of $PQ + F_1 Q.$

[asy]
unitsize(0.4 cm);

pair P, Q;
pair[] F;

path ell = yscale(sqrt(33))*xscale(7)*Circle((0,0),1);
F[1] = (4,0);
F[2] = (-4,0);
P = (0,3) + 2*dir(240);
Q = intersectionpoint(P--interp(F[2],P,5),ell);

draw(ell);
draw((-8,0)--(8,0));
draw((0,-7)--(0,7));
draw(Circle((0,3),2));
draw(F[1]--Q--F[2]);

dot("$F_1$", F[1], S);
dot("$F_2$", F[2], S);
dot("$P$", P, S);
label("$Q$", Q, NE);
[/asy]

level

Level 5

type

Intermediate Algebra

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
42d871f3-dd46-5b00-b02b-d081f87d2328
task_key
intermediate~5falgebra--test--42d871f3-dd46-5b00-b02b-d081f87d2328
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import