Chapter 8.2, Problem 3E

Solution

To calculate: The vertices and foci of the ellipse.

The vertices of the ellipse are $\left(0,6\right)$ , $\left(0,-6\right)$ and the foci are $\left(0,3\right),\left(0,-3\right)$ .

Given information:

The equation of ellipse is $\frac{y^2}{36}+\frac{x^2}{27}=1$ .

Calculation:

The vertices are the two points on the ellipse that intersect the major axis. For an ellipse with major axis parallel to the $y-$ axis, the vertices are $\left(h,k+b\right),\left(h,k-b\right)$ .

  $\frac{y^2}{36}+\frac{x^2}{27}=1$

Compare the equation with standard equation of ellipse:

Ellipse with center $\left(h,k\right)=\left(0,0\right)$ .

  $\begin{array}{l}a=\sqrt{36}=6\\ b=\sqrt{27}=3\sqrt{3}\end{array}$

Thus, the vertices of the ellipse are $\left(0,6\right)$ , $\left(0,-6\right)$ .

Calculate the value of $c:$

  $\begin{array}{l}c=\sqrt{6^2-{\left(3\sqrt{3}\right)}^2}\\ c=\sqrt{36-37}\\ c=\sqrt{9}\\ c=3\end{array}$

The foci of the ellipse are defined as $\left(h,k+c\right),\left(h,k-c\right)$ .

Thus, the foci are $\left(0,3\right),\left(0,-3\right)$ .