Chapter SH, Problem 9.11EP

To determine

To find: the equation of the ellipse shown in the graph.

Answer to Problem 9.11EP

The equation of ellipse is ${\text{81x}}^{\text{2}}+16{\text{y}}^{\text{2}}-1296=0$ .

Explanation of Solution

Given:

  

Concept used:

The equation of the ellipse if major axis at vertical will be:

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

Major axis $\text{=2a}$ .

Minor axis $\text{=2b}$ .

Calculation:

For ellipse $\text{a}$ is major axis and $\text{b}$ is minor axis.

From the given graph:

Center: $\left(\text{h,k}\right)=\left(0,0\right)$ .

Major axis is at vertical: $\text{a=9}$ .

Minor axis at horizontal: $\text{b=4}$ .

So the equation of the ellipse will be:

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

  $\frac{{\left(\text{x-0}\right)}^{\text{2}}}{4^{\text{2}}}\text{+}\frac{{\left(\text{y-0}\right)}^{\text{2}}}{9^{\text{2}}}=1$ .

  $\frac{{\text{x}}^{\text{2}}}{16}\text{+}\frac{{\text{y}}^{\text{2}}}{\text{81}}=1$ .

  $\frac{{\text{81x}}^{\text{2}}+16{\text{y}}^{\text{2}}}{16\times \text{81}}=1$ .

Hence, the equation of ellipse is ${\text{81x}}^{\text{2}}+16{\text{y}}^{\text{2}}-1296=0$ .