Chapter 15.1, Problem 39E

To determine

To find:

The limit of the area of the ellipse as c approaches $0$ .

Answer to Problem 39E

  $\pi a^2$

Explanation of Solution

Given:

The area of an ellipse with semi major axis $a$ is

  $\pi a\sqrt{a^2-c^2}$

Concept used:

General form is

  $\left(h-a,k\right)and\left(h+a,k\right)$

The general form an ellipse with horizontal major axis

  $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}$

Calculation:

The area of an ellipse with semi major axis $a$ is

  $\pi a\sqrt{a^2-c^2}\mathrm{...............}\left(1\right)$

Applying by the limit from equation (1)

  $\pi a\sqrt{a^2-c^2}$

  $\begin{array}{l}\underset{c\to 0}{\lim }A=\underset{c\to 0}{\lim }\pi a\sqrt{a^2-c^2}\\ \underset{c\to 0}{\lim }A=\pi a\sqrt{a^2-{\left(0\right)}^2}\\ \underset{c\to 0}{\lim }A=\pi a\sqrt{a^2}\\ \underset{c\to 0}{\lim }A=\pi a^2\end{array}$

The limit of the area of the ellipse as c approaches $0$ is $\pi a^2$