QJEEM22Ma5b95: JEE Main 2022 Mathematics - Class 11 JEEM22Ma5b95 Solution: Let the hyperbola and the ellipse be such that the length of…

$\because$ $H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over 1} = 1$

$\therefore$ Length of latus rectum $= {2 \over a}$

$E:{{{x^2}} \over 4} + {{{y^2}} \over 3} = 1$

Length of latus rectum $= {6 \over 2} = 3$

$\because$ ${2 \over a} = 3 \Rightarrow a = {2 \over 3}$

$\therefore$ $12\left( {e_H^2 + e_E^2} \right) = 12\left( {1 + {9 \over 4}} \right) + \left( {1 - {3 \over 4}} \right) = 42$