Question

If eccentricity of ellipse $\frac{x^2}{a^2}+\frac{y^2}{a^2+4a}=1$ is less than $\frac{1}{\sqrt{2}}$, and complete set of values of $a$ is $(-\infty ,\lambda )\cup (\mu ,\infty )$, then the value of $|\lambda +\mu |$ is

• A. $4$
• B. $8$
• C. $12$
• D. $16$

Answer 1

The correct option is A $4$

case I:

$a>0$, then $e=\sqrt{\frac{4a}{a^2+4a}}<\frac{1}{\sqrt{2}}\Rightarrow a>4$
case II:

$a<0,$ then $e=\sqrt{\frac{-4a}{a^2}}=\sqrt{\frac{-4}{a}}<\frac{1}{\sqrt{2}}\Rightarrow a<-8$
$\therefore a\in (-\infty ,-8)\cup (4,\infty )$

$\therefore |\lambda +\mu |=4$