Find the Asymptotes 64x^2+128x-9y^2-72y-656=0

$64 x^{2} + 128 x - 9 y^{2} - 72 y - 656 = 0$

Step 1

Find the standard form of the hyperbola.

Add $656$ to both sides of the equation.

$64 x^{2} + 128 x - 9 y^{2} - 72 y = 656$

Complete the square for $64 x^{2} + 128 x$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 64$

$b = 128$

$c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{128}{2 \cdot 64}$

Simplify the right side.

Cancel the common factor of $128$ and $2$ .

Factor $2$ out of $128$ .

$d = \frac{2 \cdot 64}{2 \cdot 64}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 64$ .

$d = \frac{2 \cdot 64}{2 (64)}$

Cancel the common factor.

$d = \frac{2 \cdot 64}{2 \cdot 64}$

Rewrite the expression.

$d = \frac{64}{64}$

Cancel the common factor of $64$ .

Cancel the common factor.

$d = \frac{64}{64}$

Rewrite the expression.

$d = 1$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{128^{2}}{4 \cdot 64}$

Simplify the right side.

Simplify each term.

Raise $128$ to the power of $2$ .

$e = 0 - \frac{16384}{4 \cdot 64}$

Multiply $4$ by $64$ .

$e = 0 - \frac{16384}{256}$

Divide $16384$ by $256$ .

$e = 0 - 1 \cdot 64$

Multiply $- 1$ by $64$ .

$e = 0 - 64$

Subtract $64$ from $0$ .

$e = - 64$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $64 {(x + 1)}^{2} - 64$ .

$64 {(x + 1)}^{2} - 64$

Substitute $64 {(x + 1)}^{2} - 64$ for $64 x^{2} + 128 x$ in the equation $64 x^{2} + 128 x - 9 y^{2} - 72 y = 656$ .

$64 {(x + 1)}^{2} - 64 - 9 y^{2} - 72 y = 656$

Move $- 64$ to the right side of the equation by adding $64$ to both sides.

$64 {(x + 1)}^{2} - 9 y^{2} - 72 y = 656 + 64$

Complete the square for $- 9 y^{2} - 72 y$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 9$

$b = - 72$

$c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 72}{2 \cdot - 9}$

Simplify the right side.

Cancel the common factor of $- 72$ and $2$ .

Factor $2$ out of $- 72$ .

$d = \frac{2 \cdot - 36}{2 \cdot - 9}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 9$ .

$d = \frac{2 \cdot - 36}{2 (- 9)}$

Cancel the common factor.

$d = \frac{2 \cdot - 36}{2 \cdot - 9}$

Rewrite the expression.

$d = \frac{- 36}{- 9}$

Cancel the common factor of $- 36$ and $- 9$ .

Factor $- 9$ out of $- 36$ .

$d = \frac{- 9 (4)}{- 9}$

Cancel the common factors.

Factor $- 9$ out of $- 9$ .

$d = \frac{- 9 \cdot 4}{- 9 \cdot 1}$

Cancel the common factor.

$d = \frac{- 9 \cdot 4}{- 9 \cdot 1}$

Rewrite the expression.

$d = \frac{4}{1}$

Divide $4$ by $1$ .

$d = 4$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 72)}^{2}}{4 \cdot - 9}$

Simplify the right side.

Simplify each term.

Raise $- 72$ to the power of $2$ .

$e = 0 - \frac{5184}{4 \cdot - 9}$

Multiply $4$ by $- 9$ .

$e = 0 - \frac{5184}{- 36}$

Divide $5184$ by $- 36$ .

$e = 0 - - 144$

Multiply $- 1$ by $- 144$ .

$e = 0 + 144$

Add $0$ and $144$ .

$e = 144$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 9 {(y + 4)}^{2} + 144$ .

$- 9 {(y + 4)}^{2} + 144$

Substitute $- 9 {(y + 4)}^{2} + 144$ for $- 9 y^{2} - 72 y$ in the equation $64 x^{2} + 128 x - 9 y^{2} - 72 y = 656$ .

$64 {(x + 1)}^{2} - 9 {(y + 4)}^{2} + 144 = 656 + 64$

Move $144$ to the right side of the equation by adding $144$ to both sides.

$64 {(x + 1)}^{2} - 9 {(y + 4)}^{2} = 656 + 64 - 144$

Simplify $656 + 64 - 144$ .

Add $656$ and $64$ .

$64 {(x + 1)}^{2} - 9 {(y + 4)}^{2} = 720 - 144$

Subtract $144$ from $720$ .

$64 {(x + 1)}^{2} - 9 {(y + 4)}^{2} = 576$

Divide each term by $576$ to make the right side equal to one.

$\frac{64 {(x + 1)}^{2}}{576} - \frac{9 {(y + 4)}^{2}}{576} = \frac{576}{576}$

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 $1$ .

$\frac{{(x + 1)}^{2}}{9} - \frac{{(y + 4)}^{2}}{64} = 1$

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, $a$ .

$a = 3$

$b = 8$

$k = - 4$

$h = - 1$

Step 4

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

$y = \pm \frac{8}{3} \cdot (x - (- 1)) - 4$

Step 5

Simplify to find the first asymptote.

Remove parentheses.

$y = \frac{8}{3} \cdot (x - (- 1)) - 4$

Simplify $\frac{8}{3} \cdot (x - (- 1)) - 4$ .

Simplify each term.

Multiply $- 1$ by $- 1$ .

$y = \frac{8}{3} \cdot (x + 1) - 4$

Apply the distributive property.

$y = \frac{8}{3} x + \frac{8}{3} \cdot 1 - 4$

Combine $\frac{8}{3}$ and $x$ .

$y = \frac{8 x}{3} + \frac{8}{3} \cdot 1 - 4$

Multiply $\frac{8}{3}$ by $1$ .

$y = \frac{8 x}{3} + \frac{8}{3} - 4$

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{8 x}{3} + \frac{8}{3} - 4 \cdot \frac{3}{3}$

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

$y = \frac{8 x}{3} + \frac{8}{3} + \frac{- 4 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = \frac{8 x}{3} + \frac{8 - 4 \cdot 3}{3}$

Simplify the numerator.

Multiply $- 4$ by $3$ .

$y = \frac{8 x}{3} + \frac{8 - 12}{3}$

Subtract $12$ from $8$ .

$y = \frac{8 x}{3} + \frac{- 4}{3}$

Move the negative in front of the fraction.

$y = \frac{8 x}{3} - \frac{4}{3}$

Step 6

Simplify to find the second asymptote.

Remove parentheses.

$y = - \frac{8}{3} \cdot (x - (- 1)) - 4$

Simplify $- \frac{8}{3} \cdot (x - (- 1)) - 4$ .

Simplify each term.

Multiply $- 1$ by $- 1$ .

$y = - \frac{8}{3} \cdot (x + 1) - 4$

Apply the distributive property.

$y = - \frac{8}{3} x - \frac{8}{3} \cdot 1 - 4$

Combine $x$ and $\frac{8}{3}$ .

$y = - \frac{x \cdot 8}{3} - \frac{8}{3} \cdot 1 - 4$

Multiply $- 1$ by $1$ .

$y = - \frac{x \cdot 8}{3} - \frac{8}{3} - 4$

Move $8$ to the left of $x$ .

$y = - \frac{8 x}{3} - \frac{8}{3} - 4$

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{8 x}{3} - \frac{8}{3} - 4 \cdot \frac{3}{3}$

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

$y = - \frac{8 x}{3} - \frac{8}{3} + \frac{- 4 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = - \frac{8 x}{3} + \frac{- 8 - 4 \cdot 3}{3}$

Simplify the numerator.

Multiply $- 4$ by $3$ .

$y = - \frac{8 x}{3} + \frac{- 8 - 12}{3}$

Subtract $12$ from $- 8$ .

$y = - \frac{8 x}{3} + \frac{- 20}{3}$

Move the negative in front of the fraction.

$y = - \frac{8 x}{3} - \frac{20}{3}$

Step 7

This hyperbola has two asymptotes.

$y = \frac{8 x}{3} - \frac{4}{3}, y = - \frac{8 x}{3} - \frac{20}{3}$

Step 8

The asymptotes are $y = \frac{8 x}{3} - \frac{4}{3}$ and $y = - \frac{8 x}{3} - \frac{20}{3}$ .

Asymptotes: $y = \frac{8 x}{3} - \frac{4}{3}, y = - \frac{8 x}{3} - \frac{20}{3}$

Step 9

Find the Asymptotes 64x^2+128x-9y^2-72y-656=0

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