Rectangle in Ellipse

Geometry Level 3

Find the area of the greatest rectangle that can be inscribed in in the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ .

Take $a,b> 0$ .

For more problems try my set

\sqrt ab

2 ab

\frac{a}{b}

ab

2 solutions

Otto Bretscher
Apr 4, 2016

Let me recycle an idea I have used before, showing that we don't even need calculus.

We can parameterize the ellipse as $x=a\cos(t)$ and $y=b\sin(t)$ so that the area of an inscribed rectangle is $A=4xy=4ab\cos t\sin t=2ab\sin(2t)$ for $0<t<\frac{\pi}{2}$ . The maximum of $\boxed{2ab}$ is attained when $t=\frac{\pi}{4}$ .

Good, parametrization makes the solution even faster and more elegant - how I expected from you!

Andreas Wendler - 5 years, 2 months ago

Yes. This is the shortest way :)

Pulkit Gupta - 5 years, 2 months ago

Guillermo Templado
Apr 4, 2016

You can see the solution in Lagrange multipliers .

In that page, there also is a problem asking for the greatest volume of a parallelipiped than can be inscribed in a elipsoid.

And you can generalize this problem, for example, to a finite elliptic cylinder in $\mathbb{R}^3$

You mustn't take a sledgehammer to crack a nut! Simple differential calculation is sufficient to maximize 4 x y in the ellipse.

Andreas Wendler - 5 years, 2 months ago

If you read a little bit of this article, this example can be solved with simple differential calculation.

Guillermo Templado - 5 years, 2 months ago

Simple trigonometry nails it but ;)

Pulkit Gupta - 5 years, 2 months ago

"Mit Kanonen auf Spatzen schiessen," wie wir in der Schweiz sagen ;)

Otto Bretscher - 5 years, 2 months ago

Rectangle in Ellipse

2 solutions

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