Let each of the two ellipses and have eccentricity . Let the lengths of the latus recta of and be and , respectively, such that If the distance between the foci of is 8, then the distance between the foci of is [2026]

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Solution

Q.
Let each of the two ellipses $E_{1} : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, (a > b)$ and $E_{2} : \frac{x^{2}}{A^{2}} + \frac{y^{2}}{B^{2}} = 1, (A < B)$ have eccentricity $\frac{4}{5}$ . Let the lengths of the latus recta of $E_{1}$ and $E_{2}$ be $l_{1}$ and $l_{2}$ , respectively, such that $2 l_{1}^{2} = 9 l_{2} .$ If the distance between the foci of $E_{1}$ is 8, then the distance between the foci of $E_{2}$ is [2026]

1 $\frac{8}{5}$

2 $\frac{96}{5}$

3 $\frac{16}{5}$

4 $\frac{32}{5}$

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