Chapter 8.4, Problem 4MRPS

Solution

To Find : An equation of an ellipse that has its center at the origin, a horizontal minor axis and lies completely between the given circles.

The required equation will be $\frac{x^2}{64}+\frac{y^2}{25}=1$ .

Given information :

The circles, $x^2+y^2=25$ and $x^2+y^2=64$ .

Explanation :

The standard equation of an ellipse that has its center at the origin, a horizontal minor axis is given by,

  $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$   ....... (1)

The equation of the circles are given to be,

  $x^2+y^2=25$   ....... (2)

  $x^2+y^2=64$   ....... (3)

Since the major axis is vertical, so $b^2>a^2$ .

Therefore, from (2) and (3),

  $\begin{array}{l}{b}^2=64\\ a^2=25\end{array}$

Therefore, the required equation will be $\frac{x^2}{64}+\frac{y^2}{25}=1$ .