Find the Center 25x^2+16y^2-150x+64y-111=0

$25 x^{2} + 16 y^{2} - 150 x + 64 y - 111 = 0$

Find the standard form of the ellipse.

Add $111$ to both sides of the equation.

$25 x^{2} + 16 y^{2} - 150 x + 64 y = 111$

Complete the square for $25 x^{2} - 150 x$ .

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 $25 {(x - 3)}^{2} - 225$ .

$25 {(x - 3)}^{2} - 225$

Substitute $25 {(x - 3)}^{2} - 225$ for $25 x^{2} - 150 x$ in the equation $25 x^{2} + 16 y^{2} - 150 x + 64 y = 111$ .

$25 {(x - 3)}^{2} - 225 + 16 y^{2} + 64 y = 111$

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

$25 {(x - 3)}^{2} + 16 y^{2} + 64 y = 111 + 225$

Complete the square for $16 y^{2} + 64 y$ .

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

$a = 16$

$b = 64$

$c = 0$

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

$d = \frac{64}{2 \cdot 16}$

Simplify the right side.

Cancel the common factor of $64$ and $2$ .

Factor $2$ out of $64$ .

$d = \frac{2 \cdot 32}{2 \cdot 16}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 16$ .

$d = \frac{2 \cdot 32}{2 (16)}$

Cancel the common factor.

$d = \frac{2 \cdot 32}{2 \cdot 16}$

Rewrite the expression.

$d = \frac{32}{16}$

Cancel the common factor of $32$ and $16$ .

Factor $16$ out of $32$ .

$d = \frac{16 \cdot 2}{16}$

Cancel the common factors.

Factor $16$ out of $16$ .

$d = \frac{16 \cdot 2}{16 (1)}$

Cancel the common factor.

$d = \frac{16 \cdot 2}{16 \cdot 1}$

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

$e = 0 - \frac{64^{2}}{4 \cdot 16}$

Simplify the right side.

Simplify each term.

Raise $64$ to the power of $2$ .

$e = 0 - \frac{4096}{4 \cdot 16}$

Multiply $4$ by $16$ .

$e = 0 - \frac{4096}{64}$

Divide $4096$ by $64$ .

$e = 0 - 1 \cdot 64$

Multiply $- 1$ by $64$ .

$e = 0 - 64$

Subtract $64$ from $0$ .

$e = - 64$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(y + 2)}^{2} - 64$ .

$16 {(y + 2)}^{2} - 64$

Substitute $16 {(y + 2)}^{2} - 64$ for $16 y^{2} + 64 y$ in the equation $25 x^{2} + 16 y^{2} - 150 x + 64 y = 111$ .

$25 {(x - 3)}^{2} + 16 {(y + 2)}^{2} - 64 = 111 + 225$

Move $- 64$ to the right side of the equation by adding $64$ to both sides.

$25 {(x - 3)}^{2} + 16 {(y + 2)}^{2} = 111 + 225 + 64$

Simplify $111 + 225 + 64$ .

Add $111$ and $225$ .

$25 {(x - 3)}^{2} + 16 {(y + 2)}^{2} = 336 + 64$

Add $336$ and $64$ .

$25 {(x - 3)}^{2} + 16 {(y + 2)}^{2} = 400$

Divide each term by $400$ to make the right side equal to one.

$\frac{25 {(x - 3)}^{2}}{400} + \frac{16 {(y + 2)}^{2}}{400} = \frac{400}{400}$

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 $1$ .

$\frac{{(x - 3)}^{2}}{16} + \frac{{(y + 2)}^{2}}{25} = 1$

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

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

Match the values in this ellipse to those of the standard form. The variable $a$ represents the radius of the major axis of the ellipse, $b$ represents the radius of the minor axis of the ellipse, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from the origin.

$a = 5$

$b = 4$

$k = - 2$

$h = 3$

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

$(3, - 2)$

Find the Center 25x^2+16y^2-150x+64y-111=0

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