Chapter 8.2, Problem 39E

Solution

To Find: The center, vertices, and foci of the ellipse.

Center of the ellipse $\left(7,-3\right)$ , vertices of the ellipse are $\left(7,6\right),\left(7,-12\right)$ and the foci are $\left(7,-3+\sqrt{17}\right),\left(7,-3-\sqrt{17}\right)$ .

Given information:

The equation of ellipse is $\frac{{\left(y+3\right)}^2}{81}+\frac{{\left(x-7\right)}^2}{64}=1$ .

Calculation:

The standard form of the ellipse equation is $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$ with the center $\left(h,k\right)$ and $a,b$ are the semi-major and semi-minor axes.

Rewrite $\frac{{\left(y+3\right)}^2}{81}+\frac{{\left(x-7\right)}^2}{64}=1$ in the standard ellipse equation, it follows:

  $\frac{{\left(x-7\right)}^2}{8^2}+\frac{{\left(y-\left(-3\right)\right)}^2}{9^2}=1$

Therefore, the ellipse properties are:

  $\left(h,k\right)=\left(7,-3\right),a=8,b=9$

  $b>a$ therefore $b$ is semi-major axis $a$ is semi-minor axis.

Ellipse with center $\left(h,k\right)=\left(7,-3\right),b=9,a=8$

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 $\left(h,k+b\right),\left(h,k-b\right)$ .

  $\frac{{\left(y+3\right)}^2}{81}+\frac{{\left(x-7\right)}^2}{64}=1$

Compare the equation with standard equation of ellipse:

Ellipse with center $\left(h,k\right)=\left(7,-3\right)$ .

  $\left(7,-3+9\right),\left(7,-3-9\right)=\left(7,6\right),\left(7,-12\right)$

Thus, the vertices of the ellipse are $\left(7,6\right),\left(7,-12\right)$ .

Calculate the value of $c:$

  $\begin{array}{l}c=\sqrt{{\left(9\right)}^2-{\left(8\right)}^2}\\ c=\sqrt{81-64}\\ c=\sqrt{17}\end{array}$

The foci of the ellipse are defined as $\left(h,k+c\right),\left(h,k-c\right)=\left(7,-3+\sqrt{17}\right),\left(7,-3-\sqrt{17}\right)$ .

Thus, the foci are $\left(7,-3+\sqrt{17}\right),\left(7,-3-\sqrt{17}\right)$ .