Chapter 9, Problem 1RE

In Exercises 1-8, graph each ellipse and locate the foci.

$\frac{{\text{x}}^{\text{2}}}{36}+\frac{y^2}{25}=1$

To determine

To graph: The equation of ellipse $\frac{x^2}{36}+\frac{y^2}{25}=1$ and locate the foci.

Explanation of Solution

Given information:

The ellipse equation: $\frac{x^2}{36}+\frac{y^2}{25}=1$

Graph:

Let us consider the following ellipse equation:

$\frac{x^2}{36}+\frac{y^2}{25}=1$ (I)

And the general equation of the ellipse is as given below:

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

And comparing equations (I) and (II) to obtain:

$a^2=36$ And $b^2=25$

And the equation is the general ellipse’s equation with $a^2=36$ and $b^2=25$.

Since, the denominator of the $x^2\text{-term}$ is greater than the denominator of the $y^2\text{-term}$, therefore, the major axis is horizontal.

Simplifying $a^2=36$ and $b^2=25$ as follows to obtain the value of a and b:

$\begin{array}{c}{a}^2=36\\ a^2=6^2\\ a=6\end{array}$

And

$\begin{array}{c}{b}^2=25\\ b^2=5^2\\ b=5\end{array}$

And for a standard ellipse equation, the vertices of an ellipse are $\left(–a,0\right)$ and $\left(a,0\right),$ the endpoints of the vertical minor axis are $\left(0,–b\right)$ and $\left(0,b\right)$.

So, the vertices of the ellipse are $\left(–6,0\right)$ and $\left(6,0\right)$, and the endpoints of the vertical minor axis are $\left(0,–5\right)$ and $\left(0,5\right)$.

And the foci $F_1\text{ and }{F}_2$ are located at the points $\left(–c,0\right)$ and $\left(c,0\right),$ where $c^2=a^2–b^2$.

Put the values of $a^2$ and $b^2$ in $c^2=a^2–b^2$ and find the value of $c$ as follows:

$\begin{array}{c}{c}^2=a^2–b^2\\ c^2=36-25\\ c^2=11\\ c=\sqrt{11}\end{array}$

The foci $F_1\text{ and }{F}_2$ are $\left(-\sqrt{11},0\right)$ and $\left(\sqrt{11},0\right)$.

Then, plot the endpoints and the foci and trace them to obtain a smooth curve as shown below:

Interpretation:

Thus, the foci of the provided ellipse equation are $\left(-\sqrt{11},0\right)$ and $\left(\sqrt{11},0\right)$, the vertices of the ellipse are $\left(–6,0\right)$ and $\left(6,0\right)$, and the endpoints of the vertical minor axis are $\left(0,–5\right)$ and $\left(0,5\right)$.