Find the Asymptotes ((x-3)^2)/49-((y+2)^2)/16=1

$\frac{{(x - 3)}^{2}}{49} - \frac{{(y + 2)}^{2}}{16} = 1$

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 $1$ .

$\frac{{(x - 3)}^{2}}{49} - \frac{{(y + 2)}^{2}}{16} = 1$

Step 2

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

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

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, $a$ .

$a = 7$

$b = 4$

$k = - 2$

$h = 3$

Step 4

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

$y = \pm \frac{4}{7} \cdot (x - (3)) - 2$

Step 5

Simplify to find the first asymptote.

Remove parentheses.

$y = \frac{4}{7} \cdot (x - (3)) - 2$

Simplify $\frac{4}{7} \cdot (x - (3)) - 2$ .

Simplify each term.

Multiply $- 1$ by $3$ .

$y = \frac{4}{7} \cdot (x - 3) - 2$

Apply the distributive property.

$y = \frac{4}{7} x + \frac{4}{7} \cdot - 3 - 2$

Combine $\frac{4}{7}$ and $x$ .

$y = \frac{4 x}{7} + \frac{4}{7} \cdot - 3 - 2$

Multiply $\frac{4}{7} \cdot - 3$ .

Combine $\frac{4}{7}$ and $- 3$ .

$y = \frac{4 x}{7} + \frac{4 \cdot - 3}{7} - 2$

Multiply $4$ by $- 3$ .

$y = \frac{4 x}{7} + \frac{- 12}{7} - 2$

Move the negative in front of the fraction.

$y = \frac{4 x}{7} - \frac{12}{7} - 2$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = \frac{4 x}{7} - \frac{12}{7} - 2 \cdot \frac{7}{7}$

Combine $- 2$ and $\frac{7}{7}$ .

$y = \frac{4 x}{7} - \frac{12}{7} + \frac{- 2 \cdot 7}{7}$

Combine the numerators over the common denominator.

$y = \frac{4 x}{7} + \frac{- 12 - 2 \cdot 7}{7}$

Simplify the numerator.

Multiply $- 2$ by $7$ .

$y = \frac{4 x}{7} + \frac{- 12 - 14}{7}$

Subtract $14$ from $- 12$ .

$y = \frac{4 x}{7} + \frac{- 26}{7}$

Move the negative in front of the fraction.

$y = \frac{4 x}{7} - \frac{26}{7}$

Step 6

Simplify to find the second asymptote.

Remove parentheses.

$y = - \frac{4}{7} \cdot (x - (3)) - 2$

Simplify $- \frac{4}{7} \cdot (x - (3)) - 2$ .

Simplify each term.

Multiply $- 1$ by $3$ .

$y = - \frac{4}{7} \cdot (x - 3) - 2$

Apply the distributive property.

$y = - \frac{4}{7} x - \frac{4}{7} \cdot - 3 - 2$

Combine $x$ and $\frac{4}{7}$ .

$y = - \frac{x \cdot 4}{7} - \frac{4}{7} \cdot - 3 - 2$

Multiply $- \frac{4}{7} \cdot - 3$ .

Multiply $- 3$ by $- 1$ .

$y = - \frac{x \cdot 4}{7} + 3 (\frac{4}{7}) - 2$

Combine $3$ and $\frac{4}{7}$ .

$y = - \frac{x \cdot 4}{7} + \frac{3 \cdot 4}{7} - 2$

Multiply $3$ by $4$ .

$y = - \frac{x \cdot 4}{7} + \frac{12}{7} - 2$

Move $4$ to the left of $x$ .

$y = - \frac{4 x}{7} + \frac{12}{7} - 2$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = - \frac{4 x}{7} + \frac{12}{7} - 2 \cdot \frac{7}{7}$

Combine $- 2$ and $\frac{7}{7}$ .

$y = - \frac{4 x}{7} + \frac{12}{7} + \frac{- 2 \cdot 7}{7}$

Combine the numerators over the common denominator.

$y = - \frac{4 x}{7} + \frac{12 - 2 \cdot 7}{7}$

Simplify the numerator.

Multiply $- 2$ by $7$ .

$y = - \frac{4 x}{7} + \frac{12 - 14}{7}$

Subtract $14$ from $12$ .

$y = - \frac{4 x}{7} + \frac{- 2}{7}$

Move the negative in front of the fraction.

$y = - \frac{4 x}{7} - \frac{2}{7}$

Step 7

This hyperbola has two asymptotes.

$y = \frac{4 x}{7} - \frac{26}{7}, y = - \frac{4 x}{7} - \frac{2}{7}$

Step 8

The asymptotes are $y = \frac{4 x}{7} - \frac{26}{7}$ and $y = - \frac{4 x}{7} - \frac{2}{7}$ .

Asymptotes: $y = \frac{4 x}{7} - \frac{26}{7}, y = - \frac{4 x}{7} - \frac{2}{7}$

Step 9

Find the Asymptotes ((x-3)^2)/49-((y+2)^2)/16=1

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