Chapter 12.6, Problem 1PT

To determine

The appropriate option that will complete the statement “$\frac{x^2}{25}+1=\frac{z^2}{16}-\frac{y^2}{9}$ is a(n):” from the given options.

Answer to Problem 1PT

The appropriate option is $\underset{\_}{\left(\text{b}\right)}$.

Explanation of Solution

The given equation is $\frac{x^2}{25}+1=\frac{z^2}{16}-\frac{y^2}{9}$.

It can be rewritten as follows.

$\begin{array}{l}\frac{x^2}{25}+1=\frac{z^2}{16}-\frac{y^2}{9}\\ -\frac{x^2}{5^2}-\frac{y^2}{3^2}+\frac{z^2}{4^2}=1\end{array}$

It is known that, the equation of a hyperboloid of two sheets (along z-axis) is $-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$.

Observe that $-\frac{x^2}{5^2}-\frac{y^2}{3^2}+\frac{z^2}{4^2}=1$ is in the form of $-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$.

Here, $a=5,b=3$ and $c=4$.

Thus, the equation $\frac{x^2}{25}+1=\frac{z^2}{16}-\frac{y^2}{9}$ is a hyperboloid of two sheets.

Therefore, the appropriate option is $\underset{\_}{\left(\text{b}\right)}$.