Let the ellipse E₁: x²/a²+y²/b²=1, a>b and E₂: x²/A²+y²/B²=1, A [29 Jan 2025 Shift 1]

Correct Answer is : 2465{24{6}}{5}5246

Correct Answer is : 2465{24{6}}{5}5246

Test Index

Ellipse

Section: Mathematics

Question : 27 of 117

Marks: +1, -0

Let the ellipse

E_1: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b

E_2: \frac{x^2}{A^2}+\frac{y^2}{B^2}=1, A < B

\frac{1}{\sqrt{3}}

\frac{32}{\sqrt{3}}

E_1

E_1

E_2

A, B, C

D

A B C D

Solution: 👈: Video Solution

2 a e=4

\Rightarrow a=2\sqrt{3}

\Rightarrow 1-\frac{b^2}{12}=\frac{1}{3} \Rightarrow b^2=8

\frac{2 b^2}{a} \times \frac{2 A^2}{B}=\frac{32}{\sqrt{3}}

\Rightarrow \frac{2 \times 8}{2 \sqrt{3}} \times \frac{2 A^2}{B}=\frac{32}{\sqrt{3}}

\Rightarrow \frac{A^2}{B}=2 \Rightarrow A^2=2 B

1-\frac{A^2}{B}=\frac{1}{3}

\Rightarrow B=3 \Rightarrow A^2=6

E_1: \frac{x^2}{12}+\frac{y^2}{8}=1

E_1: \frac{x^2}{6}+\frac{y^2}{9}=1

On solving (i) & (ii)

(x, y)= \left(\frac{\sqrt{6}}{\sqrt{5}}, \frac{6}{\sqrt{5}}\right), \left(\frac{-\sqrt{6}}{\sqrt{5}}, \frac{6}{\sqrt{5}}\right), \left(\frac{\sqrt{6}}{\sqrt{5}}, \frac{-6}{\sqrt{5}}\right),

\left(\frac{-\sqrt{6}}{\sqrt{5}}, \frac{-6}{\sqrt{5}}\right)

Four points are vertices of rectangle area

=\frac{24\sqrt{6}}{5}

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