Find the Asymptotes ((y-2)^2)/(1^2)-((x-4)^2)/(3^2)=1

$\frac{{(y - 2)}^{2}}{1^{2}} - \frac{{(x - 4)}^{2}}{3^{2}} = 1$

Step 1

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 $1$ .

${(y - 2)}^{2} - \frac{{(x - 4)}^{2}}{9} = 1$

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, $a$ .

$a = 1$

$b = 3$

$k = 2$

$h = 4$

Step 4

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

$y = \pm \frac{1}{3} \cdot (x - (4)) + 2$

Step 5

Simplify to find the first asymptote.

Remove parentheses.

$y = \frac{1}{3} \cdot (x - (4)) + 2$

Simplify $\frac{1}{3} \cdot (x - (4)) + 2$ .

Simplify each term.

Multiply $- 1$ by $4$ .

$y = \frac{1}{3} \cdot (x - 4) + 2$

Apply the distributive property.

$y = \frac{1}{3} x + \frac{1}{3} \cdot - 4 + 2$

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

$y = \frac{x}{3} + \frac{1}{3} \cdot - 4 + 2$

Combine $\frac{1}{3}$ and $- 4$ .

$y = \frac{x}{3} + \frac{- 4}{3} + 2$

Move the negative in front of the fraction.

$y = \frac{x}{3} - \frac{4}{3} + 2$

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{x}{3} - \frac{4}{3} + 2 \cdot \frac{3}{3}$

Combine $2$ and $\frac{3}{3}$ .

$y = \frac{x}{3} - \frac{4}{3} + \frac{2 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = \frac{x}{3} + \frac{- 4 + 2 \cdot 3}{3}$

Simplify the numerator.

Multiply $2$ by $3$ .

$y = \frac{x}{3} + \frac{- 4 + 6}{3}$

Add $- 4$ and $6$ .

$y = \frac{x}{3} + \frac{2}{3}$

Step 6

Simplify to find the second asymptote.

Remove parentheses.

$y = - \frac{1}{3} \cdot (x - (4)) + 2$

Simplify $- \frac{1}{3} \cdot (x - (4)) + 2$ .

Simplify each term.

Multiply $- 1$ by $4$ .

$y = - \frac{1}{3} \cdot (x - 4) + 2$

Apply the distributive property.

$y = - \frac{1}{3} x - \frac{1}{3} \cdot - 4 + 2$

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

$y = - \frac{x}{3} - \frac{1}{3} \cdot - 4 + 2$

Multiply $- \frac{1}{3} \cdot - 4$ .

Multiply $- 4$ by $- 1$ .

$y = - \frac{x}{3} + 4 (\frac{1}{3}) + 2$

Combine $4$ and $\frac{1}{3}$ .

$y = - \frac{x}{3} + \frac{4}{3} + 2$

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{x}{3} + \frac{4}{3} + 2 \cdot \frac{3}{3}$

Combine $2$ and $\frac{3}{3}$ .

$y = - \frac{x}{3} + \frac{4}{3} + \frac{2 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = - \frac{x}{3} + \frac{4 + 2 \cdot 3}{3}$

Simplify the numerator.

Multiply $2$ by $3$ .

$y = - \frac{x}{3} + \frac{4 + 6}{3}$

Add $4$ and $6$ .

$y = - \frac{x}{3} + \frac{10}{3}$

Step 7

This hyperbola has two asymptotes.

$y = \frac{x}{3} + \frac{2}{3}, y = - \frac{x}{3} + \frac{10}{3}$

Step 8

The asymptotes are $y = \frac{x}{3} + \frac{2}{3}$ and $y = - \frac{x}{3} + \frac{10}{3}$ .

Asymptotes: $y = \frac{x}{3} + \frac{2}{3}, y = - \frac{x}{3} + \frac{10}{3}$

Step 9

Find the Asymptotes ((y-2)^2)/(1^2)-((x-4)^2)/(3^2)=1

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