Rectangle in an Ellipse

Geometry Level 3

$ABCD$ is a rectangle whose vertices lie on the circumference of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ .

Let $P$ is the maximum area of $ABCD$ and $Q$ is the area of the ellipse. Then the value of $\frac{P}{Q}$ can be written as $\frac{n}{\pi}$ . Find $n$ .

The answer is 2.

1 solution

Ishaan Kaul
Sep 15, 2017

take a point on the ellipse as P. P=acosx,bsinx. now the rectangle is symmetrical about the diagonal. so the opp. vertex will be -acosx,-bsinx as it will be in the 3rd quad. further take the diagonal of the rec in terms of a and b you will see that length of the rec. will come 2acosx and breath will be 2bsinx area=4absinxcosx =2absin2x max. area =2ab (sin2x has max value =1) 2ab/pie a b(pie a b = area of the ellipse. so the ans is 2.

Rectangle in an Ellipse

The answer is 2.

1 solution

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