Chapter 8.4, Problem 46E

Solution

To determine : The equation of an ellipse in standard form and its graph by rewriting it as two functions.

The standard form of the equation is $\frac{x^2}{5^2}+\frac{y^2}{2^2}=1$ .

Given information :

The equation of an ellipse in standard form.

Explanation :

Consider the equation of an ellipse as:

  $4x^2+25y^2=100\mathrm{.......}\left(1\right)$

The standard equation of an ellipse with horizontal major axis and center at the origin is

  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Rewrite the equation $\left(1\right)$ .

  $4x^2+25y^2=100$

Divide both the sides by $100$ .

  $\begin{array}{l}\frac{4x^2}{100}+\frac{25y^2}{100}=1\\ \frac{x^2}{25}+\frac{y^2}{4}=1\end{array}$

Write in the standard form.

  $\frac{x^2}{5^2}+\frac{y^2}{2^2}=1$

Therefore, the standard form of the equation is $\frac{x^2}{5^2}+\frac{y^2}{2^2}=1$ .

The graph of the equation will be,