Chapter 11.3, Problem 36E

To determine

To find: the equation for the hyperbola.

Answer to Problem 36E

The equation of hyperbola is $\frac{y^2}{36}-\frac{x^2}{324}=1$ .

Explanation of Solution

Given information:

From the figure, provided in the question it is observed that

Vertices of hyperbola : $\left(0,\pm 6\right)$

asymptotes of hyperbola : $y=\pm \frac{1}{3}x$

Calculation:

since from the given vertices it is observed that the vertices of the hyperbola are on the y-axis, so the hyperbola has a vertical transverse axis.

So the equation is of the form,

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

For transverse axis hyperbola, the asymptote

  $y=\pm \frac{a}{b}x$

So from the $y=\pm \frac{1}{3}x$ asymptote it can be observed that ,

  $\frac{a}{b}=\frac{1}{3}$

Since $a=6$ , now to find $b$ substitute $a=6$ ,

  $\begin{array}{l}\frac{6}{b}=\frac{1}{3}\\ b=18\end{array}$

so the equation of hyperbola is,

  $\begin{array}{l}\frac{y^2}{{\left(a\right)}^2}-\frac{x^2}{b^2}=1\\ \frac{y^2}{{\left(6\right)}^2}-\frac{x^2}{{\left(18\right)}^2}=1\text{ [substitute]}\\ \frac{y^2}{36}-\frac{x^2}{324}=1\text{ [simplify] }\end{array}$

Hence, the equation of hyperbola is $\frac{y^2}{36}-\frac{x^2}{324}=1$ .