Find the Center y^2-x^2+10x-4y=22

$y^{2} - x^{2} + 10 x - 4 y = 22$

Step 1

Find the standard form of the hyperbola.

Complete the square for $y^{2} - 4 y$ .

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

$a = 1$

$b = - 4$

$c = 0$

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

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

Cancel the common factor of $- 4$ and $2$ .

Factor $2$ out of $- 4$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

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

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

Simplify the right side.

Simplify each term.

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

Rewrite $- 4$ as $- 1 (4)$ .

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

Apply the product rule to $- 1 (4)$ .

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

Raise $- 1$ to the power of $2$ .

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

Multiply $4^{2}$ by $1$ .

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

Factor $4$ out of $4^{2}$ .

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

Cancel the common factors.

Factor $4$ out of $4 \cdot 1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $4$ by $1$ .

$e = 0 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Subtract $4$ from $0$ .

$e = - 4$

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

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

Substitute ${(y - 2)}^{2} - 4$ for $y^{2} - 4 y$ in the equation $y^{2} - x^{2} + 10 x - 4 y = 22$ .

${(y - 2)}^{2} - 4 - x^{2} + 10 x = 22$

Move $- 4$ to the right side of the equation by adding $4$ to both sides.

${(y - 2)}^{2} - x^{2} + 10 x = 22 + 4$

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

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

Simplify the right side.

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

Factor $2$ out of $10$ .

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

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$d = - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$d = - 5$

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 $- 4$ .

$e = 0 - - 25$

Multiply $- 1$ by $- 25$ .

$e = 0 + 25$

Add $0$ and $25$ .

$e = 25$

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 $y^{2} - x^{2} + 10 x - 4 y = 22$ .

${(y - 2)}^{2} - {(x - 5)}^{2} + 25 = 22 + 4$

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

${(y - 2)}^{2} - {(x - 5)}^{2} = 22 + 4 - 25$

Simplify $22 + 4 - 25$ .

Add $22$ and $4$ .

${(y - 2)}^{2} - {(x - 5)}^{2} = 26 - 25$

Subtract $25$ from $26$ .

${(y - 2)}^{2} - {(x - 5)}^{2} = 1$

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.

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

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, $a$ .

$a = 1$

$b = 1$

$k = 2$

$h = 5$

Step 4

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

$(5, 2)$

Step 5

Find the Center y^2-x^2+10x-4y=22

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