Chapter 8.4, Problem 15E

Solution

To describe: and correct the error in the given graphing of ellipse.

Error: Major axis is not horizontal.

Correct graph:

  

Given information:

The equation of the ellipse is $\frac{x^2}{4}+\frac{y^2}{16}=1$ .

The given graph is:

  

Formula Used:

Standard equation of an Ellipse with Center at the Origin

Equation	Major Axis	Vertices	Co-vertices
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$	Horizontal	$\left(\pm a,0\right)$	$\left(0,\pm b\right)$
$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$	Vertical	$\left(0,\pm a\right)$	$\left(\pm b,0\right)$

The major and minor axes are of the lengths $2a$ and $2b$ respectively, where $a>b>0$ . The foci of the ellipse lie on the major axis at a distance of c units from the center, where $c=a^2-b^2$

Explanation:

The given equation of the ellipse is $\frac{x^2}{4}+\frac{y^2}{16}=1$ .

Here, it can be observed that the denominator of $y^2$ is greater than the denominator of $x^2$ .

So, in this case the major axis is vertical. But in the given graph, the major axis is horizontal. So, that is the error in the given graphing.

The correct graph is: