Name: Parametric Equation of an Ellipse
Uploaded: 2019-07-19
Duration: 7 min 12 s
Description: In this video, you will understand the parametric equation of an ellipse with examples, which will help you solve questions in the exam. Watch and learn

Question

An ellipse and a hyperbola have the same centre origin, the same foci and the minor-axis of the one is the same as the conjugate axis of the other. If

e_{1}, e_{2}

be their eccentricities respectively, then find value of

\frac{1}{e_{1}^{2}} + \frac{1}{e_{2}^{2}}

Accepted Answer

Let $e_{1}$ is the eccentricity of ellipse and $e_{2}$ is the eccentricity of the hyperbola.

Let ellipse $\equiv \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and hyperbola $\frac{x^{2}}{A^{2}} + \frac{y^{2}}{B^{2}} = 1$

Foci of ellipse $(\pm a e_{1}, 0)$

Foci of hyperbola $= (\pm A e_{2}, 0)$

$e_{1} = \sqrt{\frac{a^{2} - b^{2}}{a^{2}}}, e_{2} = \sqrt{\frac{A^{2} + B^{2}}{A^{2}}}$

Since the focus of the ellipse and hyperbola are same.

So, $a e_{1} = A e_{2}$

$\Rightarrow a \sqrt{\frac{a^{2} - b^{2}}{a^{2}}} = \sqrt{\frac{A^{2} + B^{2}}{A^{2}}}$

$\Rightarrow \sqrt{a^{2} - b^{2}} = \sqrt{A^{2} + B^{2}} . . . (1)$

We are given that $b = B$

So, $a^{2} - b^{2} = A^{2} + b^{2}$

$a^{2} - A^{2} = 2 b^{2}$

So, $\frac{1}{e_{1}^{2}} + \frac{1}{e_{2}^{2}} = \frac{a^{2}}{a^{2} - b^{2}} + \frac{A^{2}}{B^{2} + b^{2}}$

From equation $(1)$

$\frac{a^{2}}{A^{2} + b^{2}} + \frac{A^{2}}{A^{2} + b^{2}}$

$\Rightarrow \frac{a^{2} + A^{2}}{A^{2} + b^{2}}$

$\Rightarrow \frac{A^{2} + 2 b^{2} + A^{2}}{A^{2} + b^{2}}$

$\Rightarrow \frac{2 A^{2} + 2 b^{2}}{A^{2} + b^{2}}$

$\Rightarrow \frac{2 (A^{2} + b^{2})}{A^{2} + b^{2}}$

$= \frac{1}{e_{1}^{2}} + \frac{1}{e_{2}^{2}} = 2$

An ellipse and a hyperbola have the same centre origin, the same foci and the minor-axis of the one is the same as the conjugate axis of the other. If $e_{1}, e_{2}$ be their eccentricities respectively, then find value of $\frac{1}{e_{1}^{2}} + \frac{1}{e_{2}^{2}}$

Important Questions on Ellipse

Learn and Prepare for any exam you want !

An ellipse and a hyperbola have the same centre origin, the same foci and the minor-axis of the one is the same as the conjugate axis of the other. If e1, e2 be their eccentricities respectively, then find value of 1e12+1e22

Important Questions on Ellipse

Learn and Prepare for any exam you want !

An ellipse and a hyperbola have the same centre origin, the same foci and the minor-axis of the one is the same as the conjugate axis of the other. If $e_{1}, e_{2}$ be their eccentricities respectively, then find value of $\frac{1}{e_{1}^{2}} + \frac{1}{e_{2}^{2}}$