Chapter 10, Problem 1RE

How are the graphs of these ellipses similar and how are they different?

$\frac{x^2}{100}+\frac{y^2}{225}=1$

$\frac{{\left(x-1\right)}^2}{100}+\frac{{\left(y+7\right)}^2}{225}=1$

To determine

The difference and the similarities between the graphs of the ellipses mentioned below:

$\frac{x^2}{100}+\frac{y^2}{225}=1$

And

$\frac{{\left(x-1\right)}^2}{100}+\frac{{\left(y+7\right)}^2}{225}=1$

Answer to Problem 1RE

Both ellipses have the same minor and major axis width while they have different centers.

Explanation of Solution

Consider the equations of the ellipse:

$\frac{x^2}{100}+\frac{y^2}{225}=1$

And

$\frac{{\left(x-1\right)}^2}{100}+\frac{{\left(y+7\right)}^2}{225}=1$

The standard form of the equation of an ellipse is:

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

On comparing, the similarities between the ellipses is:

$\begin{array}{c}{a}^2=100\\ a=10\end{array}$

And,

$\begin{array}{c}{b}^2=225\\ b=15\end{array}$

Therefore, both ellipses have same minor and major axis width.

The difference between the ellipses is:

The ellipse, $\frac{x^2}{100}+\frac{y^2}{225}=1$ , has center $\left(0,0\right)$ whereas, the ellipse, $\frac{{\left(x-1\right)}^2}{100}+\frac{{\left(y+7\right)}^2}{225}=1$ , has center $\left(1,-7\right)$ .

Therefore, both the ellipses have different centers.