Question

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

Answer 1

Answer: 75

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