Chapter 8.2, Problem 7E

Solution

To Match : The graph with its equation.

The correct graph of the equation $\frac{x^2}{25}+\frac{y^2}{16}=1$ is option $\left(d\right)$ .

Given information :

The equation is $\frac{x^2}{25}+\frac{y^2}{16}=1$ .

The options of the graph are shown below:

  

Explanation :

The standard form of the ellipse equation is $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$ with the center $\left(h,k\right)$ and $a,b$ are the semi-major and semi-minor axes.

Rewrite $\frac{x^2}{25}+\frac{y^2}{16}=1$ in the standard ellipse equation, it follows:

  $\frac{{\left(x-0\right)}^2}{5^2}+\frac{{\left(y-0\right)}^2}{4^2}=1$

Therefore, the ellipse properties are:

  $\left(h,k\right)=\left(0,0\right),a=5,b=4$

  $a>b$ therefore $a$ is semi-major axis $b$ is semi-minor axis.

Ellipse with center $\left(h,k\right)=0\text{,}a=5,b=4$

As the given tick marks on all axes are $1$ unit apart. The equation $\frac{x^2}{25}+\frac{y^2}{16}=1$ is matching with the option $\left(d\right)$ .