If e1 and e2 are the eccentricities of the ellipse
Question:

If $e_{1}$ and $e_{2}$ are the eccentricities of the ellipse, $\frac{x^{2}}{18}+\frac{y^{2}}{4}=1$

and the hyperbola, $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$ respectively and $\left(e_{1}, e_{2}\right)$ is

a point on the ellipse, $15 x^{2}+3 y^{2}=k$, then $k$ is equal to

  1. (1) 16

  2. (2) 17

  3. (3) 15

  4. (4) 14


Correct Option: 1

Solution:

Eccentricity of ellipse

$e_{1}=\sqrt{1-\frac{4}{18}}=\sqrt{\frac{7}{9}}=\frac{\sqrt{7}}{3}$

Eccentricity of hyperbola

$e_{2}=\sqrt{1+\frac{4}{9}}=\sqrt{\frac{13}{9}}=\frac{\sqrt{13}}{3}$

Since, the point $\left(e_{1}, e_{2}\right)$ is on the ellipse

$15 x^{2}+3 y^{2}=k$

Then, $15 e_{1}^{2}+3 e_{2}^{2}=k$

$\Rightarrow \quad k=15\left(\frac{7}{9}\right)+3\left(\frac{13}{9}\right)$

$\Rightarrow \quad k=16$

Administrator

Leave a comment

Please enter comment.
Please enter your name.