Find the Center 11x^2-25y^2+22x+250y-889=0

$11 x^{2} - 25 y^{2} + 22 x + 250 y - 889 = 0$

Find the standard form of the hyperbola.

Add $889$ to both sides of the equation.

$11 x^{2} - 25 y^{2} + 22 x + 250 y = 889$

Complete the square for $11 x^{2} + 22 x$ .

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

$a = 11$

$b = 22$

$c = 0$

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

$d = \frac{22}{2 \cdot 11}$

Simplify the right side.

Cancel the common factor of $22$ and $2$ .

Factor $2$ out of $22$ .

$d = \frac{2 \cdot 11}{2 \cdot 11}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 11$ .

$d = \frac{2 \cdot 11}{2 (11)}$

Cancel the common factor.

$d = \frac{2 \cdot 11}{2 \cdot 11}$

Rewrite the expression.

$d = \frac{11}{11}$

Cancel the common factor of $11$ .

Cancel the common factor.

$d = \frac{11}{11}$

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

$e = 0 - \frac{22^{2}}{4 \cdot 11}$

Simplify the right side.

Simplify each term.

Raise $22$ to the power of $2$ .

$e = 0 - \frac{484}{4 \cdot 11}$

Multiply $4$ by $11$ .

$e = 0 - \frac{484}{44}$

Divide $484$ by $44$ .

$e = 0 - 1 \cdot 11$

Multiply $- 1$ by $11$ .

$e = 0 - 11$

Subtract $11$ from $0$ .

$e = - 11$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $11 {(x + 1)}^{2} - 11$ .

$11 {(x + 1)}^{2} - 11$

Substitute $11 {(x + 1)}^{2} - 11$ for $11 x^{2} + 22 x$ in the equation $11 x^{2} - 25 y^{2} + 22 x + 250 y = 889$ .

$11 {(x + 1)}^{2} - 11 - 25 y^{2} + 250 y = 889$

Move $- 11$ to the right side of the equation by adding $11$ to both sides.

$11 {(x + 1)}^{2} - 25 y^{2} + 250 y = 889 + 11$

Complete the square for $- 25 y^{2} + 250 y$ .

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

$a = - 25$

$b = 250$

$c = 0$

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

$d = \frac{250}{2 \cdot - 25}$

Simplify the right side.

Cancel the common factor of $250$ and $2$ .

Factor $2$ out of $250$ .

$d = \frac{2 \cdot 125}{2 \cdot - 25}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 25$ .

$d = \frac{2 \cdot 125}{2 (- 25)}$

Cancel the common factor.

$d = \frac{2 \cdot 125}{2 \cdot - 25}$

Rewrite the expression.

$d = \frac{125}{- 25}$

Cancel the common factor of $125$ and $- 25$ .

Factor $25$ out of $125$ .

$d = \frac{25 (5)}{- 25}$

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

$e = 0 - \frac{250^{2}}{4 \cdot - 25}$

Simplify the right side.

Simplify each term.

Raise $250$ to the power of $2$ .

$e = 0 - \frac{62500}{4 \cdot - 25}$

Multiply $4$ by $- 25$ .

$e = 0 - \frac{62500}{- 100}$

Divide $62500$ by $- 100$ .

$e = 0 - - 625$

Multiply $- 1$ by $- 625$ .

$e = 0 + 625$

Add $0$ and $625$ .

$e = 625$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 25 {(y - 5)}^{2} + 625$ .

$- 25 {(y - 5)}^{2} + 625$

Substitute $- 25 {(y - 5)}^{2} + 625$ for $- 25 y^{2} + 250 y$ in the equation $11 x^{2} - 25 y^{2} + 22 x + 250 y = 889$ .

$11 {(x + 1)}^{2} - 25 {(y - 5)}^{2} + 625 = 889 + 11$

Move $625$ to the right side of the equation by adding $625$ to both sides.

$11 {(x + 1)}^{2} - 25 {(y - 5)}^{2} = 889 + 11 - 625$

Simplify $889 + 11 - 625$ .

Add $889$ and $11$ .

$11 {(x + 1)}^{2} - 25 {(y - 5)}^{2} = 900 - 625$

Subtract $625$ from $900$ .

$11 {(x + 1)}^{2} - 25 {(y - 5)}^{2} = 275$

Divide each term by $275$ to make the right side equal to one.

$\frac{11 {(x + 1)}^{2}}{275} - \frac{25 {(y - 5)}^{2}}{275} = \frac{275}{275}$

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 $1$ .

$\frac{{(x + 1)}^{2}}{25} - \frac{{(y - 5)}^{2}}{11} = 1$

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

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

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, $a$ .

$a = 5$

$b = \sqrt{11}$

$k = 5$

$h = - 1$

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

$(- 1, 5)$

Find the Center 11x^2-25y^2+22x+250y-889=0

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