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If e(1)ande(2) be the eccentricities of ...

If `e_(1)ande_(2)` be the eccentricities of the ellipses `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1and(x^(2))/(a^(2))+(4y^(2))/(b^(2))=1` respectively then prove that `3=4e_(2)^(2)-e_(1)^(2)`.

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To solve the problem, we need to find the eccentricities \( e_1 \) and \( e_2 \) of the given ellipses and then prove that \( 3 = 4e_2^2 - e_1^2 \). ### Step 1: Identify the equations of the ellipses The equations of the ellipses given are: 1. \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) (Ellipse 1) 2. \( \frac{x^2}{a^2} + \frac{4y^2}{b^2} = 1 \) (Ellipse 2) ...
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