Question

The ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and hyperbola $\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$ are having a same foci and length of minor axis of ellipse is same as the conjugate axis of the hyperbola. If $e_1$ & $e_2$ are the eccentricities of ellipse and hyperbola respectively, then the value of $\frac{1}{e_1^2}+\frac{1}{e_2^2}$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

Since, ellipse and hyperbola has the same foci, therefore the major axis of ellipse will be $x-$ axis.
Now, $a{e}_1=A{e}_2;b=B$
$\Rightarrow e_1^2{a}^2=A^2{e}_2^2$
$\Rightarrow e_1^2\cdot \frac{b^2}{1-e_1^2}=e_2^2\cdot \frac{B^2}{e_2^2-1}$
$\Rightarrow \frac{1}{\frac{1}{e_1^2}-1}=\frac{1}{1-\frac{1}{e_2^2}}$
$\Rightarrow \frac{1}{e_1^2}+\frac{1}{e_2^2}=2$