Find the Center 9x^2-y^2-72x+8y+119=0

$9 x^{2} - y^{2} - 72 x + 8 y + 119 = 0$

Step 1

Find the standard form of the hyperbola.

Subtract $119$ from both sides of the equation.

$9 x^{2} - y^{2} - 72 x + 8 y = - 119$

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

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 $- 36$ .

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

Cancel the common factors.

Factor $9$ out of $9$ .

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

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 $36$ .

$e = 0 - 1 \cdot 144$

Multiply $- 1$ by $144$ .

$e = 0 - 144$

Subtract $144$ from $0$ .

$e = - 144$

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

$9 {(x - 4)}^{2} - 144$

Substitute $9 {(x - 4)}^{2} - 144$ for $9 x^{2} - 72 x$ in the equation $9 x^{2} - y^{2} - 72 x + 8 y = - 119$ .

$9 {(x - 4)}^{2} - 144 - y^{2} + 8 y = - 119$

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

$9 {(x - 4)}^{2} - y^{2} + 8 y = - 119 + 144$

Complete the square for $- y^{2} + 8 y$ .

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

$a = - 1$

$b = 8$

$c = 0$

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

$d = \frac{8}{2 \cdot - 1}$

Simplify the right side.

Cancel the common factor of $8$ and $2$ .

Factor $2$ out of $8$ .

$d = \frac{2 \cdot 4}{2 \cdot - 1}$

Move the negative one from the denominator of $\frac{4}{- 1}$ .

$d = - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$d = - 4$

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

$e = 0 - \frac{8^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

Raise $8$ to the power of $2$ .

$e = 0 - \frac{64}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

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

Divide $64$ by $- 4$ .

$e = 0 - - 16$

Multiply $- 1$ by $- 16$ .

$e = 0 + 16$

Add $0$ and $16$ .

$e = 16$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(y - 4)}^{2} + 16$ .

$- {(y - 4)}^{2} + 16$

Substitute $- {(y - 4)}^{2} + 16$ for $- y^{2} + 8 y$ in the equation $9 x^{2} - y^{2} - 72 x + 8 y = - 119$ .

$9 {(x - 4)}^{2} - {(y - 4)}^{2} + 16 = - 119 + 144$

Move $16$ to the right side of the equation by adding $16$ to both sides.

$9 {(x - 4)}^{2} - {(y - 4)}^{2} = - 119 + 144 - 16$

Simplify $- 119 + 144 - 16$ .

Add $- 119$ and $144$ .

$9 {(x - 4)}^{2} - {(y - 4)}^{2} = 25 - 16$

Subtract $16$ from $25$ .

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

Divide each term by $9$ to make the right side equal to one.

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

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 $1$ .

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

Step 2

This is the form of a hyperbola. Use this form to determine the values used to find vertices and 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, $a$ .

$a = 1$

$b = 3$

$k = 4$

$h = 4$

Step 4

The center of a hyperbola follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(4, 4)$

Step 5

Find the Center 9x^2-y^2-72x+8y+119=0

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