Find the Asymptotes x^2-9y^2-10x-36y-36=0

$x^{2} - 9 y^{2} - 10 x - 36 y - 36 = 0$

Step 1

Find the standard form of the hyperbola.

Add $36$ to both sides of the equation.

$x^{2} - 9 y^{2} - 10 x - 36 y = 36$

Complete the square for $x^{2} - 10 x$ .

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

$a = 1$

$b = - 10$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 10}{2 \cdot 1}$

Cancel the common factor of $- 10$ and $2$ .

Factor $2$ out of $- 10$ .

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 5}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 5}{1}$

Divide $- 5$ by $1$ .

$d = - 5$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 10)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 10$ to the power of $2$ .

$e = 0 - \frac{100}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{100}{4}$

Divide $100$ by $4$ .

$e = 0 - 1 \cdot 25$

Multiply $- 1$ by $25$ .

$e = 0 - 25$

Subtract $25$ from $0$ .

$e = - 25$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 5)}^{2} - 25$ .

${(x - 5)}^{2} - 25$

Substitute ${(x - 5)}^{2} - 25$ for $x^{2} - 10 x$ in the equation $x^{2} - 9 y^{2} - 10 x - 36 y = 36$ .

${(x - 5)}^{2} - 25 - 9 y^{2} - 36 y = 36$

Move $- 25$ to the right side of the equation by adding $25$ to both sides.

${(x - 5)}^{2} - 9 y^{2} - 36 y = 36 + 25$

Complete the square for $- 9 y^{2} - 36 y$ .

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

$a = - 9$

$b = - 36$

$c = 0$

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 $d = \frac{b}{2 a}$ .

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

Simplify the right side.

Cancel the common factor of $- 36$ and $2$ .

Factor $2$ out of $- 36$ .

$d = \frac{2 \cdot - 18}{2 \cdot - 9}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 18}{2 (- 9)}$

Cancel the common factor.

$d = \frac{2 \cdot - 18}{2 \cdot - 9}$

Rewrite the expression.

$d = \frac{- 18}{- 9}$

Cancel the common factor of $- 18$ and $- 9$ .

Factor $- 9$ out of $- 18$ .

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

Cancel the common factors.

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

$d = \frac{- 9 \cdot 2}{- 9 \cdot 1}$

Cancel the common factor.

$d = \frac{- 9 \cdot 2}{- 9 \cdot 1}$

Rewrite the expression.

$d = \frac{2}{1}$

Divide $2$ by $1$ .

$d = 2$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 36)}^{2}}{4 \cdot - 9}$

Simplify the right side.

Simplify each term.

Raise $- 36$ to the power of $2$ .

$e = 0 - \frac{1296}{4 \cdot - 9}$

Multiply $4$ by $- 9$ .

$e = 0 - \frac{1296}{- 36}$

Divide $1296$ by $- 36$ .

$e = 0 - - 36$

Multiply $- 1$ by $- 36$ .

$e = 0 + 36$

Add $0$ and $36$ .

$e = 36$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 9 {(y + 2)}^{2} + 36$ .

$- 9 {(y + 2)}^{2} + 36$

Substitute $- 9 {(y + 2)}^{2} + 36$ for $- 9 y^{2} - 36 y$ in the equation $x^{2} - 9 y^{2} - 10 x - 36 y = 36$ .

${(x - 5)}^{2} - 9 {(y + 2)}^{2} + 36 = 36 + 25$

Move $36$ to the right side of the equation by adding $36$ to both sides.

${(x - 5)}^{2} - 9 {(y + 2)}^{2} = 36 + 25 - 36$

Simplify $36 + 25 - 36$ .

Add $36$ and $25$ .

${(x - 5)}^{2} - 9 {(y + 2)}^{2} = 61 - 36$

Subtract $36$ from $61$ .

${(x - 5)}^{2} - 9 {(y + 2)}^{2} = 25$

Divide each term by $25$ to make the right side equal to one.

$\frac{{(x - 5)}^{2}}{25} - \frac{9 {(y + 2)}^{2}}{25} = \frac{25}{25}$

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 $1$ .

$\frac{{(x - 5)}^{2}}{25} - \frac{{(y + 2)}^{2}}{\frac{25}{9}} = 1$

Step 2

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

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

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, $a$ .

$a = 5$

$b = \frac{5}{3}$

$k = - 2$

$h = 5$

Step 4

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

$y = \pm \frac{1}{3} \cdot (x - (5)) - 2$

Step 5

Simplify to find the first asymptote.

Remove parentheses.

$y = \frac{1}{3} \cdot (x - (5)) - 2$

Simplify $\frac{1}{3} \cdot (x - (5)) - 2$ .

Simplify each term.

Multiply $- 1$ by $5$ .

$y = \frac{1}{3} \cdot (x - 5) - 2$

Apply the distributive property.

$y = \frac{1}{3} x + \frac{1}{3} \cdot - 5 - 2$

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

$y = \frac{x}{3} + \frac{1}{3} \cdot - 5 - 2$

Combine $\frac{1}{3}$ and $- 5$ .

$y = \frac{x}{3} + \frac{- 5}{3} - 2$

Move the negative in front of the fraction.

$y = \frac{x}{3} - \frac{5}{3} - 2$

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

$y = \frac{x}{3} - \frac{5}{3} - 2 \cdot \frac{3}{3}$

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

$y = \frac{x}{3} - \frac{5}{3} + \frac{- 2 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = \frac{x}{3} + \frac{- 5 - 2 \cdot 3}{3}$

Simplify the numerator.

Multiply $- 2$ by $3$ .

$y = \frac{x}{3} + \frac{- 5 - 6}{3}$

Subtract $6$ from $- 5$ .

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

Move the negative in front of the fraction.

$y = \frac{x}{3} - \frac{11}{3}$

Step 6

Simplify to find the second asymptote.

Remove parentheses.

$y = - \frac{1}{3} \cdot (x - (5)) - 2$

Simplify $- \frac{1}{3} \cdot (x - (5)) - 2$ .

Simplify each term.

Multiply $- 1$ by $5$ .

$y = - \frac{1}{3} \cdot (x - 5) - 2$

Apply the distributive property.

$y = - \frac{1}{3} x - \frac{1}{3} \cdot - 5 - 2$

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

$y = - \frac{x}{3} - \frac{1}{3} \cdot - 5 - 2$

Multiply $- \frac{1}{3} \cdot - 5$ .

Multiply $- 5$ by $- 1$ .

$y = - \frac{x}{3} + 5 (\frac{1}{3}) - 2$

Combine $5$ and $\frac{1}{3}$ .

$y = - \frac{x}{3} + \frac{5}{3} - 2$

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

$y = - \frac{x}{3} + \frac{5}{3} - 2 \cdot \frac{3}{3}$

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

$y = - \frac{x}{3} + \frac{5}{3} + \frac{- 2 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = - \frac{x}{3} + \frac{5 - 2 \cdot 3}{3}$

Simplify the numerator.

Multiply $- 2$ by $3$ .

$y = - \frac{x}{3} + \frac{5 - 6}{3}$

Subtract $6$ from $5$ .

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

Move the negative in front of the fraction.

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

Step 7

This hyperbola has two asymptotes.

$y = \frac{x}{3} - \frac{11}{3}, y = - \frac{x}{3} - \frac{1}{3}$

Step 8

The asymptotes are $y = \frac{x}{3} - \frac{11}{3}$ and $y = - \frac{x}{3} - \frac{1}{3}$ .

Asymptotes: $y = \frac{x}{3} - \frac{11}{3}, y = - \frac{x}{3} - \frac{1}{3}$

Step 9

Find the Asymptotes x^2-9y^2-10x-36y-36=0

Related Questions

Support LCMGCF.com by disabling your adblocker.