Find the Center 9x^2+25y^2-90x+200y+400=0

$9 x^{2} + 25 y^{2} - 90 x + 200 y + 400 = 0$

Step 1

Find the standard form of the ellipse.

Subtract $400$ from both sides of the equation.

$9 x^{2} + 25 y^{2} - 90 x + 200 y = - 400$

Complete the square for $9 x^{2} - 90 x$ .

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

$a = 9$

$b = - 90$

$c = 0$

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

$d = \frac{- 90}{2 \cdot 9}$

Simplify the right side.

Cancel the common factor of $- 90$ and $2$ .

Factor $2$ out of $- 90$ .

$d = \frac{2 \cdot - 45}{2 \cdot 9}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 45}{2 (9)}$

Cancel the common factor.

$d = \frac{2 \cdot - 45}{2 \cdot 9}$

Rewrite the expression.

$d = \frac{- 45}{9}$

Cancel the common factor of $- 45$ and $9$ .

Factor $9$ out of $- 45$ .

$d = \frac{9 \cdot - 5}{9}$

Cancel the common factors.

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 5}{9 (1)}$

Cancel the common factor.

$d = \frac{9 \cdot - 5}{9 \cdot 1}$

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

$e = 0 - \frac{{(- 90)}^{2}}{4 \cdot 9}$

Simplify the right side.

Simplify each term.

Raise $- 90$ to the power of $2$ .

$e = 0 - \frac{8100}{4 \cdot 9}$

Multiply $4$ by $9$ .

$e = 0 - \frac{8100}{36}$

Divide $8100$ by $36$ .

$e = 0 - 1 \cdot 225$

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 $9 {(x - 5)}^{2} - 225$ .

$9 {(x - 5)}^{2} - 225$

Substitute $9 {(x - 5)}^{2} - 225$ for $9 x^{2} - 90 x$ in the equation $9 x^{2} + 25 y^{2} - 90 x + 200 y = - 400$ .

$9 {(x - 5)}^{2} - 225 + 25 y^{2} + 200 y = - 400$

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

$9 {(x - 5)}^{2} + 25 y^{2} + 200 y = - 400 + 225$

Complete the square for $25 y^{2} + 200 y$ .

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

$a = 25$

$b = 200$

$c = 0$

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

$d = \frac{200}{2 \cdot 25}$

Simplify the right side.

Cancel the common factor of $200$ and $2$ .

Factor $2$ out of $200$ .

$d = \frac{2 \cdot 100}{2 \cdot 25}$

Cancel the common factors.

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

$d = \frac{2 \cdot 100}{2 (25)}$

Cancel the common factor.

$d = \frac{2 \cdot 100}{2 \cdot 25}$

Rewrite the expression.

$d = \frac{100}{25}$

Cancel the common factor of $100$ and $25$ .

Factor $25$ out of $100$ .

$d = \frac{25 \cdot 4}{25}$

Cancel the common factors.

Factor $25$ out of $25$ .

$d = \frac{25 \cdot 4}{25 (1)}$

Cancel the common factor.

$d = \frac{25 \cdot 4}{25 \cdot 1}$

Rewrite the expression.

$d = \frac{4}{1}$

Divide $4$ by $1$ .

$d = 4$

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

$e = 0 - \frac{200^{2}}{4 \cdot 25}$

Simplify the right side.

Simplify each term.

Raise $200$ to the power of $2$ .

$e = 0 - \frac{40000}{4 \cdot 25}$

Multiply $4$ by $25$ .

$e = 0 - \frac{40000}{100}$

Divide $40000$ by $100$ .

$e = 0 - 1 \cdot 400$

Multiply $- 1$ by $400$ .

$e = 0 - 400$

Subtract $400$ from $0$ .

$e = - 400$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $25 {(y + 4)}^{2} - 400$ .

$25 {(y + 4)}^{2} - 400$

Substitute $25 {(y + 4)}^{2} - 400$ for $25 y^{2} + 200 y$ in the equation $9 x^{2} + 25 y^{2} - 90 x + 200 y = - 400$ .

$9 {(x - 5)}^{2} + 25 {(y + 4)}^{2} - 400 = - 400 + 225$

Move $- 400$ to the right side of the equation by adding $400$ to both sides.

$9 {(x - 5)}^{2} + 25 {(y + 4)}^{2} = - 400 + 225 + 400$

Simplify $- 400 + 225 + 400$ .

Add $- 400$ and $225$ .

$9 {(x - 5)}^{2} + 25 {(y + 4)}^{2} = - 175 + 400$

Add $- 175$ and $400$ .

$9 {(x - 5)}^{2} + 25 {(y + 4)}^{2} = 225$

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

$\frac{9 {(x - 5)}^{2}}{225} + \frac{25 {(y + 4)}^{2}}{225} = \frac{225}{225}$

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 $1$ .

$\frac{{(x - 5)}^{2}}{25} + \frac{{(y + 4)}^{2}}{9} = 1$

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.

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

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.

$a = 5$

$b = 3$

$k = - 4$

$h = 5$

Step 4

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

$(5, - 4)$

Step 5

Find the Center 9x^2+25y^2-90x+200y+400=0

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