Question

If the length of the latus rectum of the ellipse $x ^2+4 y ^2+2 x +8 y -\lambda=0$ is 4 , and $l$ is the length of its major axis, then $\lambda+l$ is equal to______

Answer 1

$\lambda+\ell=75 $
$ x^2+4 y^2+2 x+8 y-\lambda=0$
$ \frac{(x+1)^2}{\lambda+5}+\frac{(y+1)^2}{\frac{\lambda+5}{4}}=1 $
$\because \frac{2 b^2}{a}=4$
$ \frac{2(\lambda+5)}{4}=4(\sqrt{\lambda+5}) $
$ \Rightarrow \lambda=59 $
$\lambda \neq-5$
$ l=2 a=2 \sqrt{\lambda+5}=2 \sqrt{65}=16$
$ \Rightarrow \lambda+\ell=59+16=75$