QJEEM22Ma7224: JEE Main 2022 Mathematics - Class 11 JEEM22Ma7224 Solution: If the length of the latus rectum of the ellipse is 4 , and is…

Equation of ellipse is : ${x^2} + 4{y^2} + 2x + 8y - \lambda = 0$

${(x + 1)^2} + 4{(y + 1)^2} = \lambda + 5$

${{{{(x + 1)}^2}} \over {\lambda + 5}} + {{{{(y + 1)}^2}} \over {\left( {{{\lambda + 5} \over 4}} \right)}} = 1$

Length of latus rectum $= {{2\,.\,\left( {{{\lambda + 5} \over 4}} \right)} \over {\sqrt {\lambda + 5} }} = 4$ .

$\therefore$ $\lambda = 59$ .

Length of major axis $= 2\,.\,\sqrt {\lambda + 5} = 16 = l$

$\therefore$ $\lambda + l = 75$ .