Chapter 10.3, Problem 27AYU

In Problems 27-38, find an equation for each ellipse. Graph the equation by hand.

Center at $\left(0,0\right)$; focus at $\left(3,0\right)$; vertex at $\left(5,0\right)$

To determine

(a & b)

To find: An equation for each ellipse.

Answer to Problem 27AYU

Solution:

$\frac{x^2}{25}+\frac{y^2}{16}=1$

Explanation of Solution

Given:

Center at $\left(0,0\right)$, focus at $\left(3,0\right)$, Vertex at $\left(5,0\right)$.

Formula used:

Center	Major axis	Foci	Vertices	Equation
$\left(0,0\right)$	$x\text{-axis}$	$\left(c,0\right)$	$\left(a,0\right)$	$\begin{array}{l}x²/a²+y²/b²=1\\ a>b>0\text{and}b²=a²-c²\end{array}$
$\left(0,0\right)$	$x\text{-axis}$	$\left(-\text{c},0\right)$	$\left(-a,0\right)$

Calculation:

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

Focus: $\left(3,0\right)=\left(c,0\right);c=3$.

Vertex: $\left(5,0\right)=\left(a,0\right);a=5$.

$b²=a²-c²$

$b^2=5^2-3^2=25-9=16$

$b=4$

Major axis is $x\text{-axis}$.

Equation of the ellipse is given by,

$\frac{x^2}{a²}+\frac{y^2}{b²}=1$

$\frac{x^2}{25}+\frac{y^2}{16}=1$