Chapter 9.2, Problem 74E

To determine

To write the equation of the circle having the same area as the ellipse.

Answer to Problem 74E

The equation of the circle is $x^2+y^2=80$ .

Explanation of Solution

Given information:

The given statement is:

“Write an equation of the circle centered at the origin that has the same area as the ellipse represented by the equation $\frac{{\left(x-4\right)}^2}{100}+\frac{{\left(y-1\right)}^2}{64}=1$ .”

Formula used:

The following formula is used:

Area of the ellipse $A=\pi ab$

Area of the circle $=\pi r^2$

Calculation:

From the equation of the ellipse $a=10\text{ and }b=8$ .

Calculate the area of the ellipse:

  $\begin{array}{l}A=\pi ab\\ A=\pi \cdot 10\cdot 8\\ A=80\pi \end{array}$

The area of the circle is $80\pi$ .

Calculate the radius of the circle:

  $\begin{array}{l}\pi r^2=80\pi \\ r^2=80\end{array}$

The standard form of the equation of the circle is:

  ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$

Since the circle is centered at the origin.

The equation of the circle is $x^2+y^2=80$ .