Chapter 8.4, Problem 45E

Solution

To graph : The original equation and the revised equations in the same coordinate plane.

Given information :

Consider the graph of $\frac{x^2}{9}+\frac{y^2}{25}=1$ . The denominator of the $y^2-$ term first changes from $25\text{ to }9$ and then from $9\text{ to }4$ .

Graph :

Consider the given equation $\frac{x^2}{9}+\frac{y^2}{25}=1$ .

In the given equation, the denominator of $x^2-$ term is less than $y^2-$ term, this means that the major axis of the ellipse is vertical. So, the ellipse will be vertical.

The vertices of the ellipse will be $\left(3,0\right),\left(-3,0\right),\left(0,5\right),\text{ and}\left(0,-5\right)$ .

Draw the ellipse.

  

Now, as the denominator of the $y^2-$ term first changes from $25\text{ to }9$ , the equation will be $\frac{x^2}{9}+\frac{y^2}{9}=1$ , this can be written as $x^2+y^2=9$ .

This will be a circle with radius $3$ .

Draw the circle in the same coordinate plane.

  

Now, as the denominator of the $y^2-$ term first changes from $9\text{ to }4$ , the equation will be $\frac{x^2}{9}+\frac{y^2}{4}=1$ . In the given equation, the denominator of $y^2-$ term is less than $x^2-$ term, this means that the major axis of the ellipse is horizontal. So, the ellipse will be horizontal.

The vertices of the ellipse will be $\left(3,0\right),\left(-3,0\right),\left(0,2\right),\text{ and}\left(0,-2\right)$ .

Draw the ellipse in the same coordinate plane.

  

Interpretation :

The vertices of the original ellipse will be $\left(3,0\right),\left(-3,0\right),\left(0,5\right),\text{ and}\left(0,-5\right)$ , after the first change it will be a circle with radius $3$ and after the second change it will be horizontal ellipse with vertices $\left(3,0\right),\left(-3,0\right),\left(0,2\right),\text{ and}\left(0,-2\right)$ .