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MATH / The graph of x²−2xy+3y²=5 is an ellipse, although its axes are not parallel to the coordinate axes. Two horizontal lines and two vertical lines lie tangent to the ellipse, forming …

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problem

The graph of x2−2xy+3y2=5x^2 - 2xy + 3y^2 = 5 is an ellipse, although its axes are not parallel to the coordinate axes. Two horizontal lines and two vertical lines lie tangent to the ellipse, forming a rectangle, as shown: [asy] size(7cm); draw(rotate(20)*xscale(2.4)*unitcircle); draw((-3.5,0)--(3.5,0),EndArrow); draw((0,-2.5)--(0,2.5),EndArrow); real r1=2.29; draw((r1,-2.2)--(r1,2.2),dotted); draw((-r1,-2.2)--(-r1,2.2),dotted); real r2=1.26; draw((-3,r2)--(3,r2),dotted); draw((-3,-r2)--(3,-r2),dotted); label("xx",(3.5,0),E); label("yy",(0,2.5),N); [/asy] What is the area of the rectangle?
Plain-text mathematical notation (without MathML)
The graph of x²−2xy+3y²=5 is an ellipse, although its axes are not parallel to the coordinate axes. Two horizontal lines and two vertical lines lie tangent to the ellipse, forming a rectangle, as shown:
[asy]
size(7cm);
draw(rotate(20)*xscale(2.4)*unitcircle);
draw((-3.5,0)--(3.5,0),EndArrow);
draw((0,-2.5)--(0,2.5),EndArrow);
real r1=2.29; draw((r1,-2.2)--(r1,2.2),dotted);
draw((-r1,-2.2)--(-r1,2.2),dotted);
real r2=1.26; draw((-3,r2)--(3,r2),dotted);
draw((-3,-r2)--(3,-r2),dotted);
label("x",(3.5,0),E); label("y",(0,2.5),N);
[/asy]
What is the area of the rectangle?
Original LaTeX notation
The graph of $x^2 - 2xy + 3y^2 = 5$ is an ellipse, although its axes are not parallel to the coordinate axes. Two horizontal lines and two vertical lines lie tangent to the ellipse, forming a rectangle, as shown:
[asy]
size(7cm);
draw(rotate(20)*xscale(2.4)*unitcircle);
draw((-3.5,0)--(3.5,0),EndArrow);
draw((0,-2.5)--(0,2.5),EndArrow);
real r1=2.29; draw((r1,-2.2)--(r1,2.2),dotted);
draw((-r1,-2.2)--(-r1,2.2),dotted);
real r2=1.26; draw((-3,r2)--(3,r2),dotted);
draw((-3,-r2)--(3,-r2),dotted);
label("$x$",(3.5,0),E); label("$y$",(0,2.5),N);
[/asy]
What is the area of the rectangle?

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
bccb8507-699c-59f0-a181-5e734c6b4d1b
task_key
intermediate~5falgebra--test--bccb8507-699c-59f0-a181-5e734c6b4d1b
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import