Chapter 10.3, Problem 57AYU

In Problems 55-64, find an equation for each ellipse. Graph the equation by hand.

Center at $\left(4,3\right)$; vertex at $\left(4,9\right)$; focus at $\left(4,8\right)$

To determine

(a & b)

To find: An equation for each ellipse.

Answer to Problem 57AYU

Solution:

$\frac{{\left(x-4\right)}^2}{5}+\frac{{\left(y-6\right)}^2}{9}=1$

Explanation of Solution

Given:

Vertices at $\left(4,3\right)\text{and}\left(4,9\right)$; focus at $\left(4,8\right)$.

Formula used:

Center	Major axis	Foci	Vertices	Equation
$\left(h,k\right)$	$\text{parallel}\text{to}y\text{-axis}$	$\left(h,k+c\right)$	$\left(h,k+a\right)$	$\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1$ $a>b>0\text{and}b²=a²-c²$
$\left(h,k\right)$	$\text{parallel}\text{to}y\text{-axis}$	$\left(h,k-c\right)$	$\left(h,k-a\right)$

Calculation:

The center of the ellipse $\left(h,k\right)$.

Vertices: $\left(4,3\right)=\left(h,k-a\right);\left(4,9\right)=\left(h,k+a\right);h=4;k=6;a=3$.

Focus: $\left(4,8\right)=\left(h,k+c\right);c=2$.

$b²=a²-c²$

$b^2=3^2-2^2=5$

$b=\surd 5$

Center: $\left(4,6\right)$.

Major axis is parallel to $y\text{-axis}$.

Equation of the ellipse is given by,

$\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1$

$\frac{{\left(x-4\right)}^2}{5}+\frac{{\left(y-6\right)}^2}{9}=1$