Find the Asymptotes 9y^2-x^2=81

$9 y^{2} - x^{2} = 81$

Step 1

Find the standard form of the hyperbola.

Divide each term by $81$ to make the right side equal to one.

$\frac{9 y^{2}}{81} - \frac{x^{2}}{81} = \frac{81}{81}$

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{y^{2}}{9} - \frac{x^{2}}{81} = 1$

Step 2

This is the form of a hyperbola. Use this form to determine the values used to find the asymptotes of the hyperbola.

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

Step 3

Match the values in this hyperbola to those of the standard form. The variable $h$ represents the x-offset from the origin, $k$ represents the y-offset from origin, $a$ .

$a = 3$

$b = 9$

$k = 0$

$h = 0$

Step 4

The asymptotes follow the form $y = \pm \frac{a (x - h)}{b} + k$ because this hyperbola opens up and down.

$y = \pm \frac{1}{3} x + 0$

Step 5

Simplify $\frac{1}{3} x + 0$ .

Add $\frac{1}{3} x$ and $0$ .

$y = \frac{1}{3} x$

Combine $\frac{1}{3}$ and $x$ .

$y = \frac{x}{3}$

Step 6

Simplify $- \frac{1}{3} x + 0$ .

Add $- \frac{1}{3} x$ and $0$ .

$y = - \frac{1}{3} x$

Combine $x$ and $\frac{1}{3}$ .

$y = - \frac{x}{3}$

Step 7

This hyperbola has two asymptotes.

$y = \frac{x}{3}, y = - \frac{x}{3}$

Step 8

The asymptotes are $y = \frac{x}{3}$ and $y = - \frac{x}{3}$ .

Asymptotes: $y = \frac{x}{3}, y = - \frac{x}{3}$

Step 9

Find the Asymptotes 9y^2-x^2=81

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