Chapter 11.4, Problem 5E

To determine

To find: the center, foci and vertices of the ellipse and determine its major and minor axis, also sketch the graph

Answer to Problem 5E

The Center $=\left(2,1\right)$ , Foci $=\left(2\pm \sqrt{5},1\right)$ , Vertex $=\left(-1,1\right)\&\left(5,1\right)$ , Major axis $=6$ , Minor axis $=4$

Explanation of Solution

Given information: The question is $\frac{{\left(x-2\right)}^2}{9}+\frac{{\left(y-1\right)}^2}{4}=1$

Calculation: for the center and foci of ellipse

  $\begin{array}{l}{b}^2=a^2\left(1-e^2\right)\\ a^2=9;b^2=4\\ 4=9\left(1-e^2\right)\\ \frac{4}{9}=1-e^2\\ e^2=1-\frac{4}{9}\Rightarrow \frac{9-4}{9}\Rightarrow \frac{5}{9}\\ e=\sqrt{\frac{5}{9}}\Rightarrow \frac{\sqrt{5}}{3}\end{array}$

So,

Center $=\left(2,1\right)$

Foci $=\left(2\pm \sqrt{5},1\right)$

Vertex $=\left(-1,1\right)\&\left(5,1\right)$

Major axis $=6$

Minor axis $=4$