Question

If $e_1$ and $e_2$ are respectively the eccentricities of the ellipse $\frac{x^2}{18}+\frac{y^2}{4}=1$ and the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$, then the relation between $e_1$ and $e_2$ is:

• A. ${{3e}_1}^2+{e_2}^2=2$
• B. ${e_1}^2+{{2e}_2}^2=3$
• C. ${{2e}_1}^2+{e_2}^2=3$
• D. ${e_1}^2+{{3e}_2}^2=2$

Answer 1

Answer: (C)

${{2e}_1}^2+{e_2}^2=3$

For ellipse $a^2=18,b^2=4$
$\Rightarrow e_1=\sqrt{1-\frac{4}{18}}=\frac{\sqrt{7}}{3}$
For hyperbola $a^2=9,b^2=4$
$\Rightarrow e_2=\sqrt{1+\frac{4}{9}}=\frac{\sqrt{13}}{3}$
Clearly, $2e_1^2+e_2^2=3$
Hence, option 'C' is correct.