Chapter 10.3, Problem 39E

To determine

To write: The equation of the ellipse that has the center at the origin, $a=2\text{ and }e=\frac{3}{4}$

Answer to Problem 39E

  $\frac{x^2}{4}+\frac{y^2}{\left(\frac{7}{4}\right)}=1\text{ or }\frac{x^2}{\left(\frac{7}{4}\right)}+\frac{y^2}{4}=1$

Explanation of Solution

Given information: The center at the origin, $a=2\text{ and }e=\frac{3}{4}$

Calculation:

Center at origin starts us at:

  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

If $a=2\text{ and }e=\frac{c}{2}=\frac{3}{4}\to c=\frac{3}{2}$

If $a^2-b^2=c^2\to b^2=4-\frac{9}{4}=\frac{7}{4}$

There isn’t enough information to indicate a major or minor axis direction. So we have 2 possible results.

This gives us finally:

  $\frac{x^2}{4}+\frac{y^2}{\left(\frac{7}{4}\right)}=1\text{ or }\frac{x^2}{\left(\frac{7}{4}\right)}+\frac{y^2}{4}=1$