Chapter 10.3, Problem 19AYU

In problems 17-26, find the vertices and foci of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility.

$\frac{x^2}{9}+\frac{y^2}{25}=1$

To determine

(a & b)

To find: The vertices and foci of the given equation of the ellipse.

Answer to Problem 19AYU

Solution:

Vertices: $\left(0,5\right)$ and $\left(0,-5\right)$.

Foci: $\left(0, 4\right)$ and $\left(0,-4\right)$.

Explanation of Solution

Given:

$\frac{x^2}{9}+\frac{y^2}{25}=1$

Formula Used:

Equation	Center	Major axis	Foci	Vertices
$\begin{array}{l}\frac{x^2}{b^2}+\frac{y^2}{a^2}=1\\ $ a>b>0\text{ and }{b}^2=a^2-c^2$\end{array}$	$\left(0,0\right)$	$y\text{-axis}$	$\left(0,\mathrm{c}\right)$	$\left(0,\mathrm{a}\right)$
	$\left(0,0\right)$	$y\text{-axis}$	$\left(0,-c\right)$

Calculation:

From the given equation $\frac{x^2}{9}+\frac{y^2}{25}=1$.

The center of the ellipse is at origin $\left(0,0\right)$;

$a=\sqrt{25}=5$; $b=\sqrt{9}=3$.

The vertices are $\left(0, \pm \mathrm{a}\right)$, hence vertices:$\left(0,5\right)$ and $\left(0,-5\right)$.

To find the value of $c$,

$c²=\mathrm{25}²-3²=25-9=16;c=4$

Hence the foci are $\left(0,4\right)$ and $\left(0,-4\right)$.

To determine

c.

To graph: The equation $\frac{x^2}{9}+\frac{y^2}{25}=1$.

Answer to Problem 19AYU

Solution:

Explanation of Solution

Given:

$\frac{x^2}{9}+\frac{y^2}{25}=1$

Calculation:

The graph of the equation of $\frac{x^2}{9}+\frac{y^2}{25}=1$ plotted using graphing tool is shown below.

Using the given information $\frac{x^2}{9}+\frac{y^2}{25}=1$, we find the $x\text{-intercepts}$ and $y\text{-intercepts}$.

$x\text{-intercepts}$ $\left(3, 0\right) \text{and} \left(-3,0\right)$

$y\text{-intercepts}$ $\left(0,5\right)$ and $\left(-0,5\right)$

Using these points given above, we can plot the points and draw a graph. We find the center $\left(0,0\right)$, focus $\left(0,\pm 4\right)$ and vertex $\left(0,\pm 5\right)$, We see that the major axis is $y\text{-axis}$. We plot the graph for the equation using the graphing tool to verify it.