Chapter 7, Problem 1RE

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

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

To determine

The explanation how are the graphs of the ellipses $\frac{x^2}{100}+\frac{y^2}{225}=1$ and $\frac{{\left(x-1\right)}^2}{100}+\frac{{\left(y+7\right)}^2}{225}=1$ same and different.

Answer to Problem 1RE

Solution:

Each equation represents an ellipse with a vertical major axis of length 30 units and horizontal minor axis of length 20 units. However, the first equation represents an ellipse centered at $\left(0,0\right)$, whereas the second equation represents an ellipse centered at $\left(1,-7\right)$.

Explanation of Solution

Given Information:

The equations of ellipse are $\frac{x^2}{100}+\frac{y^2}{225}=1$ and $\frac{{\left(x-1\right)}^2}{100}+\frac{{\left(y+7\right)}^2}{225}=1$.

Use the formula,

If in the ellipse, the major axis is $y$-axis, the equation of the ellipse is given by:

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

Where; the center is $\left(h,k\right)$, the length of major axis is $2a$ and the length of minor axis is $2b$.

Consider the provided ellipses:

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

The First Ellipse $\frac{x^2}{100}+\frac{y^2}{225}=1$ in standard form can be written as:

$\frac{{\left(x-0\right)}^2}{{\left(10\right)}^2}+\frac{{\left(y-0\right)}^2}{{\left(15\right)}^2}=1$

So, the ellipse has its center at $\left(0,0\right)$.

Length of the major axis is $2\left(15\right)=30$ units and the length of minor axis is $2\left(10\right)=20$ units.

The Second Ellipse:

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

The ellipse has its center at $\left(1,-7\right)$

Length of the major axis is $2\left(15\right)=30$ units and the length of minor axis is $2\left(10\right)=20$ units.

The graphs of the equations are sketched as:

Each equation represents an ellipse with a vertical major axis of length $30$ units and horizontal minor axis of length $20$ units. However, the first equation represents an ellipse centered at $\left(0,0\right)$, whereas the second equation represents an ellipse centered at $\left(1,-7\right)$.