Short Answer

Prove that for an ellipse or a hyperbola the eccentricity is given by
$e = \frac{the distance between the foci}{the distance between the vertices}$

Hence proved.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

We are given,

$e = \frac{the distance between the foci}{the distance between the vertices}$

Step 2. Explanation.

Now,

$Let the foci of the ellipse are (\pm \sqrt{A^{2} - B^{2}}, 0) . Formula for the distance = \sqrt{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2}} Distance = \sqrt{{(\sqrt{A^{2} - B^{2}} - (- \sqrt{A^{2} - B^{2}}))}^{2} + (0 - 0)^{2}} Distance = \sqrt{{(2 \sqrt{A^{2} - B^{2}})}^{2}} Thus the distance between the foci is, Distance = 2 \sqrt{A^{2} - B^{2}} W e h a v e t h e e c c e n t r i c i t y e = \frac{\sqrt{A^{2} - B^{2}}}{A} Then e = \frac{2 \sqrt{A^{2} - B^{2}}}{2 A} Thus, e = \frac{the distance between foci}{the distance between the vertices}$

Hence proved.

Find Study Materials for

Create Study Materials

Select your language

Calculus

Answers without the blur.

Short Answer

Prove that for an ellipse or a hyperbola the eccentricity is given by
$e = \frac{the distance between the foci}{the distance between the vertices}$

Step by Step Solution

Step 1. Given information.

Step 2. Explanation.

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Calculus

Answers without the blur.

Short Answer

Prove that for an ellipse or a hyperbola the eccentricity is given by e= the distance between the foci the distance between the vertices

Step by Step Solution

Step 1. Given information.

Step 2. Explanation.

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Prove that for an ellipse or a hyperbola the eccentricity is given by
$e = \frac{the distance between the foci}{the distance between the vertices}$