Find the Asymptotes ((y+8)^2)/121-((x-3)^2)/25=1

$\frac{{(y + 8)}^{2}}{121} - \frac{{(x - 3)}^{2}}{25} = 1$

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 $1$ .

$\frac{{(y + 8)}^{2}}{121} - \frac{{(x - 3)}^{2}}{25} = 1$

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, $a$ .

$a = 11$

$b = 5$

$k = - 8$

$h = 3$

Step 4

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

$y = \pm \frac{11}{5} \cdot (x - (3)) - 8$

Step 5

Simplify to find the first asymptote.

Remove parentheses.

$y = \frac{11}{5} \cdot (x - (3)) - 8$

Simplify $\frac{11}{5} \cdot (x - (3)) - 8$ .

Simplify each term.

Multiply $- 1$ by $3$ .

$y = \frac{11}{5} \cdot (x - 3) - 8$

Apply the distributive property.

$y = \frac{11}{5} x + \frac{11}{5} \cdot - 3 - 8$

Combine $\frac{11}{5}$ and $x$ .

$y = \frac{11 x}{5} + \frac{11}{5} \cdot - 3 - 8$

Multiply $\frac{11}{5} \cdot - 3$ .

Combine $\frac{11}{5}$ and $- 3$ .

$y = \frac{11 x}{5} + \frac{11 \cdot - 3}{5} - 8$

Multiply $11$ by $- 3$ .

$y = \frac{11 x}{5} + \frac{- 33}{5} - 8$

Move the negative in front of the fraction.

$y = \frac{11 x}{5} - \frac{33}{5} - 8$

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{11 x}{5} - \frac{33}{5} - 8 \cdot \frac{5}{5}$

Combine $- 8$ and $\frac{5}{5}$ .

$y = \frac{11 x}{5} - \frac{33}{5} + \frac{- 8 \cdot 5}{5}$

Combine the numerators over the common denominator.

$y = \frac{11 x}{5} + \frac{- 33 - 8 \cdot 5}{5}$

Simplify the numerator.

Multiply $- 8$ by $5$ .

$y = \frac{11 x}{5} + \frac{- 33 - 40}{5}$

Subtract $40$ from $- 33$ .

$y = \frac{11 x}{5} + \frac{- 73}{5}$

Move the negative in front of the fraction.

$y = \frac{11 x}{5} - \frac{73}{5}$

Step 6

Simplify to find the second asymptote.

Remove parentheses.

$y = - \frac{11}{5} \cdot (x - (3)) - 8$

Simplify $- \frac{11}{5} \cdot (x - (3)) - 8$ .

Simplify each term.

Multiply $- 1$ by $3$ .

$y = - \frac{11}{5} \cdot (x - 3) - 8$

Apply the distributive property.

$y = - \frac{11}{5} x - \frac{11}{5} \cdot - 3 - 8$

Combine $x$ and $\frac{11}{5}$ .

$y = - \frac{x \cdot 11}{5} - \frac{11}{5} \cdot - 3 - 8$

Multiply $- \frac{11}{5} \cdot - 3$ .

Multiply $- 3$ by $- 1$ .

$y = - \frac{x \cdot 11}{5} + 3 (\frac{11}{5}) - 8$

Combine $3$ and $\frac{11}{5}$ .

$y = - \frac{x \cdot 11}{5} + \frac{3 \cdot 11}{5} - 8$

Multiply $3$ by $11$ .

$y = - \frac{x \cdot 11}{5} + \frac{33}{5} - 8$

Move $11$ to the left of $x$ .

$y = - \frac{11 x}{5} + \frac{33}{5} - 8$

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = - \frac{11 x}{5} + \frac{33}{5} - 8 \cdot \frac{5}{5}$

Combine $- 8$ and $\frac{5}{5}$ .

$y = - \frac{11 x}{5} + \frac{33}{5} + \frac{- 8 \cdot 5}{5}$

Combine the numerators over the common denominator.

$y = - \frac{11 x}{5} + \frac{33 - 8 \cdot 5}{5}$

Simplify the numerator.

Multiply $- 8$ by $5$ .

$y = - \frac{11 x}{5} + \frac{33 - 40}{5}$

Subtract $40$ from $33$ .

$y = - \frac{11 x}{5} + \frac{- 7}{5}$

Move the negative in front of the fraction.

$y = - \frac{11 x}{5} - \frac{7}{5}$

Step 7

This hyperbola has two asymptotes.

$y = \frac{11 x}{5} - \frac{73}{5}, y = - \frac{11 x}{5} - \frac{7}{5}$

Step 8

The asymptotes are $y = \frac{11 x}{5} - \frac{73}{5}$ and $y = - \frac{11 x}{5} - \frac{7}{5}$ .

Asymptotes: $y = \frac{11 x}{5} - \frac{73}{5}, y = - \frac{11 x}{5} - \frac{7}{5}$

Step 9

Find the Asymptotes ((y+8)^2)/121-((x-3)^2)/25=1

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