Find the Center 25x^2+9y^2-250x+36y+436=0

$25 x^{2} + 9 y^{2} - 250 x + 36 y + 436 = 0$

Step 1

Find the standard form of the ellipse.

Subtract $436$ from both sides of the equation.

$25 x^{2} + 9 y^{2} - 250 x + 36 y = - 436$

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

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 $- 125$ .

$d = \frac{25 \cdot - 5}{25}$

Cancel the common factors.

Factor $25$ out of $25$ .

$d = \frac{25 \cdot - 5}{25 (1)}$

Cancel the common factor.

$d = \frac{25 \cdot - 5}{25 \cdot 1}$

Rewrite the expression.

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

Divide $- 5$ by $1$ .

$d = - 5$

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 $100$ .

$e = 0 - 1 \cdot 625$

Multiply $- 1$ by $625$ .

$e = 0 - 625$

Subtract $625$ from $0$ .

$e = - 625$

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

$25 {(x - 5)}^{2} - 625$

Substitute $25 {(x - 5)}^{2} - 625$ for $25 x^{2} - 250 x$ in the equation $25 x^{2} + 9 y^{2} - 250 x + 36 y = - 436$ .

$25 {(x - 5)}^{2} - 625 + 9 y^{2} + 36 y = - 436$

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

$25 {(x - 5)}^{2} + 9 y^{2} + 36 y = - 436 + 625$

Complete the square for $9 y^{2} + 36 y$ .

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

$a = 9$

$b = 36$

$c = 0$

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

$d = \frac{36}{2 \cdot 9}$

Simplify the right side.

Cancel the common factor of $36$ and $2$ .

Factor $2$ out of $36$ .

$d = \frac{2 \cdot 18}{2 \cdot 9}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 9$ .

$d = \frac{2 \cdot 18}{2 (9)}$

Cancel the common factor.

$d = \frac{2 \cdot 18}{2 \cdot 9}$

Rewrite the expression.

$d = \frac{18}{9}$

Cancel the common factor of $18$ and $9$ .

Factor $9$ out of $18$ .

$d = \frac{9 \cdot 2}{9}$

Cancel the common factors.

Factor $9$ out of $9$ .

$d = \frac{9 \cdot 2}{9 (1)}$

Cancel the common factor.

$d = \frac{9 \cdot 2}{9 \cdot 1}$

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

$e = 0 - \frac{36^{2}}{4 \cdot 9}$

Simplify the right side.

Simplify each term.

Raise $36$ to the power of $2$ .

$e = 0 - \frac{1296}{4 \cdot 9}$

Multiply $4$ by $9$ .

$e = 0 - \frac{1296}{36}$

Divide $1296$ by $36$ .

$e = 0 - 1 \cdot 36$

Multiply $- 1$ by $36$ .

$e = 0 - 36$

Subtract $36$ from $0$ .

$e = - 36$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(y + 2)}^{2} - 36$ .

$9 {(y + 2)}^{2} - 36$

Substitute $9 {(y + 2)}^{2} - 36$ for $9 y^{2} + 36 y$ in the equation $25 x^{2} + 9 y^{2} - 250 x + 36 y = - 436$ .

$25 {(x - 5)}^{2} + 9 {(y + 2)}^{2} - 36 = - 436 + 625$

Move $- 36$ to the right side of the equation by adding $36$ to both sides.

$25 {(x - 5)}^{2} + 9 {(y + 2)}^{2} = - 436 + 625 + 36$

Simplify $- 436 + 625 + 36$ .

Add $- 436$ and $625$ .

$25 {(x - 5)}^{2} + 9 {(y + 2)}^{2} = 189 + 36$

Add $189$ and $36$ .

$25 {(x - 5)}^{2} + 9 {(y + 2)}^{2} = 225$

Divide each term by $225$ to make the right side equal to one.

$\frac{25 {(x - 5)}^{2}}{225} + \frac{9 {(y + 2)}^{2}}{225} = \frac{225}{225}$

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 $1$ .

$\frac{{(x - 5)}^{2}}{9} + \frac{{(y + 2)}^{2}}{25} = 1$

Step 2

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.

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

Step 3

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.

$a = 5$

$b = 3$

$k = - 2$

$h = 5$

Step 4

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

$(5, - 2)$

Step 5

Find the Center 25x^2+9y^2-250x+36y+436=0

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