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10-2. Parabola, Ellipse, Hyperbola
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Let $E_1: \frac{x^2}{9}+\frac{y^2}{4}=1$ be an ellipse. Ellipses $E_i$ 's are constructed such that their centres and eccentricities are same as that of $E _1$, and the length of minor axis of $E _{ i }$ is the length of major axis of $E _{ i +1}( i \geq 1)$. If $A _{ i }$ is the area of the ellipse $E _{ i }$, then $\frac{5}{\pi}\left(\sum_{ i =1}^{\infty} A _{ i }\right)$, is equal to _____
A$54$
B$55$
C$56$
D$57$
(JEE MAIN-2025)
Solution
image$E_1=\frac{x^2}{9}+\frac{y^2}{4} \Rightarrow e=\sqrt{1-\frac{4}{9}}=\frac{\sqrt{5}}{3}$
$E_2: \frac{x^2}{a^2}+\frac{y^2}{4}=1$
$e=\frac{\sqrt{5}}{3}=\sqrt{1-\frac{a^2}{4}} \Rightarrow \frac{5}{9}=1-\frac{a^2}{4}$
$ a ^2=\frac{16}{9}$
$E _2: \frac{ x ^2}{\frac{16}{9}}+\frac{ y ^2}{4}=1$
$E _3: \frac{ x ^2}{\frac{16}{9}}+\frac{ y ^2}{b^2}=1$
$e =\frac{\sqrt{5}}{3}=\sqrt{1-\frac{ b ^2}{\frac{16}{9}}} \Rightarrow b^2=\frac{64}{81}$
$ E _3=\frac{ x ^2}{\frac{16}{9}}+\frac{ y ^2}{\frac{64}{81}}=1$
$A_1=\pi \times 3 \times 2 \Rightarrow 6 \pi$
$A_2=\pi \times \frac{4}{3} \times 2=\frac{8 \pi}{3}$
$A_3=\pi \times \frac{4}{3} \times \frac{8}{9}=\frac{32 \pi}{81}$
$\sum_{ i =1}^{\infty} A _{ i }=6 \pi+\frac{8 \pi}{3}+\frac{32 \pi}{81}+\ldots \infty \Rightarrow \frac{6 \pi}{1-\frac{4}{9}} \Rightarrow \frac{54 \pi}{5}$
$\therefore \frac{5}{\pi} \sum_{ i =1}^{\infty} A _{ i } \Rightarrow \frac{5}{\pi} \times \frac{54 \pi}{5}=54$
Standard 11
Mathematics
