Chapter 9.2, Problem 70E

To determine

To find: The equation of an ellipse such that for any point on the ellipse, the sum of the distances from the points $\left(2,2\right)$ and $\left(10,2\right)$ is $36$ .

Answer to Problem 70E

The equation of the ellipse is $\frac{{\left(x-6\right)}^2}{324}+\frac{{\left(y-2\right)}^2}{308}=1$ .

Explanation of Solution

Given information:

The sum of the distances from the points $\left(2,2\right)$ and $\left(10,2\right)$ to any point on the ellipse is $36$ .

Calculation:

By definition, an ellipse is the set of all points in a plane, the sum of whose distances from the two fixed points in the plane is a constant.

Hence, $\left(2,2\right)$ and $\left(10,2\right)$ are the foci of ellipse that is ${\text{F}}_1=\left(2,2\right){\text{ and F}}_2=\left(10,2\right)$ .

The location of foci of the ellipse implies that it has a horizontal major axis so that the standard equation is:

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

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

The center is the midpoint of foci and hence it is calculated as:

  $\begin{array}{l}\left(h,k\right)=\left(\frac{10+2}{2},\frac{2+2}{2}\right)\\ =\left(\frac{12}{2},\frac{4}{2}\right)\\ =\left(6,2\right)\end{array}$

The length of the major axis is $2a$ and the sum of distance from a point on the ellipse to foci is also $2a$ .

  $\begin{array}{l}2a=36\\ a=18\end{array}$

Calculate distance of a focus that is ${\text{F}}_1=\left(2,2\right)$ to the center of the ellipse.

  $\begin{array}{l}c=\sqrt{{\left(6-2\right)}^2+{\left(2-2\right)}^2}\\ c=4\end{array}$

Use the relation $a^2=b^2+c^2$ to calculate $b^2$ .

  $\begin{array}{l}{\left(18\right)}^2=b^2+4^2\\ b^2=324-16\\ b^2=308\end{array}$

The equation of ellipse is given by:

  $\begin{array}{l}\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1\\ \frac{{\left(x-6\right)}^2}{324}+\frac{{\left(y-2\right)}^2}{308}=1\end{array}$

Therefore, the equation of the ellipse is $\frac{{\left(x-6\right)}^2}{324}+\frac{{\left(y-2\right)}^2}{308}=1$ .