Find the Center 25y^2-144x^2+150y-576x-3951=0

$25 y^{2} - 144 x^{2} + 150 y - 576 x - 3951 = 0$

Step 1

Find the standard form of the hyperbola.

Add $3951$ to both sides of the equation.

$25 y^{2} - 144 x^{2} + 150 y - 576 x = 3951$

Complete the square for $25 y^{2} + 150 y$ .

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

$a = 25$

$b = 150$

$c = 0$

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

$d = \frac{150}{2 \cdot 25}$

Simplify the right side.

Cancel the common factor of $150$ and $2$ .

Factor $2$ out of $150$ .

$d = \frac{2 \cdot 75}{2 \cdot 25}$

Cancel the common factors.

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

$d = \frac{2 \cdot 75}{2 (25)}$

Cancel the common factor.

$d = \frac{2 \cdot 75}{2 \cdot 25}$

Rewrite the expression.

$d = \frac{75}{25}$

Cancel the common factor of $75$ and $25$ .

Factor $25$ out of $75$ .

$d = \frac{25 \cdot 3}{25}$

Cancel the common factors.

Factor $25$ out of $25$ .

$d = \frac{25 \cdot 3}{25 (1)}$

Cancel the common factor.

$d = \frac{25 \cdot 3}{25 \cdot 1}$

Rewrite the expression.

$d = \frac{3}{1}$

Divide $3$ by $1$ .

$d = 3$

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

$e = 0 - \frac{150^{2}}{4 \cdot 25}$

Simplify the right side.

Simplify each term.

Raise $150$ to the power of $2$ .

$e = 0 - \frac{22500}{4 \cdot 25}$

Multiply $4$ by $25$ .

$e = 0 - \frac{22500}{100}$

Divide $22500$ by $100$ .

$e = 0 - 1 \cdot 225$

Multiply $- 1$ by $225$ .

$e = 0 - 225$

Subtract $225$ from $0$ .

$e = - 225$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $25 {(y + 3)}^{2} - 225$ .

$25 {(y + 3)}^{2} - 225$

Substitute $25 {(y + 3)}^{2} - 225$ for $25 y^{2} + 150 y$ in the equation $25 y^{2} - 144 x^{2} + 150 y - 576 x = 3951$ .

$25 {(y + 3)}^{2} - 225 - 144 x^{2} - 576 x = 3951$

Move $- 225$ to the right side of the equation by adding $225$ to both sides.

$25 {(y + 3)}^{2} - 144 x^{2} - 576 x = 3951 + 225$

Complete the square for $- 144 x^{2} - 576 x$ .

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

$a = - 144$

$b = - 576$

$c = 0$

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

$d = \frac{- 576}{2 \cdot - 144}$

Simplify the right side.

Cancel the common factor of $- 576$ and $2$ .

Factor $2$ out of $- 576$ .

$d = \frac{2 \cdot - 288}{2 \cdot - 144}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 144$ .

$d = \frac{2 \cdot - 288}{2 (- 144)}$

Cancel the common factor.

$d = \frac{2 \cdot - 288}{2 \cdot - 144}$

Rewrite the expression.

$d = \frac{- 288}{- 144}$

Cancel the common factor of $- 288$ and $- 144$ .

Factor $- 144$ out of $- 288$ .

$d = \frac{- 144 (2)}{- 144}$

Cancel the common factors.

Factor $- 144$ out of $- 144$ .

$d = \frac{- 144 \cdot 2}{- 144 \cdot 1}$

Cancel the common factor.

$d = \frac{- 144 \cdot 2}{- 144 \cdot 1}$

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

$e = 0 - \frac{{(- 576)}^{2}}{4 \cdot - 144}$

Simplify the right side.

Simplify each term.

Raise $- 576$ to the power of $2$ .

$e = 0 - \frac{331776}{4 \cdot - 144}$

Multiply $4$ by $- 144$ .

$e = 0 - \frac{331776}{- 576}$

Divide $331776$ by $- 576$ .

$e = 0 - - 576$

Multiply $- 1$ by $- 576$ .

$e = 0 + 576$

Add $0$ and $576$ .

$e = 576$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 144 {(x + 2)}^{2} + 576$ .

$- 144 {(x + 2)}^{2} + 576$

Substitute $- 144 {(x + 2)}^{2} + 576$ for $- 144 x^{2} - 576 x$ in the equation $25 y^{2} - 144 x^{2} + 150 y - 576 x = 3951$ .

$25 {(y + 3)}^{2} - 144 {(x + 2)}^{2} + 576 = 3951 + 225$

Move $576$ to the right side of the equation by adding $576$ to both sides.

$25 {(y + 3)}^{2} - 144 {(x + 2)}^{2} = 3951 + 225 - 576$

Simplify $3951 + 225 - 576$ .

Add $3951$ and $225$ .

$25 {(y + 3)}^{2} - 144 {(x + 2)}^{2} = 4176 - 576$

Subtract $576$ from $4176$ .

$25 {(y + 3)}^{2} - 144 {(x + 2)}^{2} = 3600$

Divide each term by $3600$ to make the right side equal to one.

$\frac{25 {(y + 3)}^{2}}{3600} - \frac{144 {(x + 2)}^{2}}{3600} = \frac{3600}{3600}$

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 $1$ .

$\frac{{(y + 3)}^{2}}{144} - \frac{{(x + 2)}^{2}}{25} = 1$

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, $a$ .

$a = 12$

$b = 5$

$k = - 3$

$h = - 2$

Step 4

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

$(- 2, - 3)$

Step 5

Find the Center 25y^2-144x^2+150y-576x-3951=0

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