Find the Center x^2+2x+9y^2-36y+1=0

$x^{2} + 2 x + 9 y^{2} - 36 y + 1 = 0$

Step 1

Find the standard form of the ellipse.

Subtract $1$ from both sides of the equation.

$x^{2} + 2 x + 9 y^{2} - 36 y = - 1$

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

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

$a = 1$

$b = 2$

$c = 0$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

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

$e = 0 - \frac{2^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{4}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{4}{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$e = 0 - \frac{4}{4}$

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Subtract $1$ from $0$ .

$e = - 1$

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

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

Substitute ${(x + 1)}^{2} - 1$ for $x^{2} + 2 x$ in the equation $x^{2} + 2 x + 9 y^{2} - 36 y = - 1$ .

${(x + 1)}^{2} - 1 + 9 y^{2} - 36 y = - 1$

Move $- 1$ to the right side of the equation by adding $1$ to both sides.

${(x + 1)}^{2} + 9 y^{2} - 36 y = - 1 + 1$

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 $x^{2} + 2 x + 9 y^{2} - 36 y = - 1$ .

${(x + 1)}^{2} + 9 {(y - 2)}^{2} - 36 = - 1 + 1$

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

${(x + 1)}^{2} + 9 {(y - 2)}^{2} = - 1 + 1 + 36$

Simplify $- 1 + 1 + 36$ .

Add $- 1$ and $1$ .

${(x + 1)}^{2} + 9 {(y - 2)}^{2} = 0 + 36$

Add $0$ and $36$ .

${(x + 1)}^{2} + 9 {(y - 2)}^{2} = 36$

Divide each term by $36$ to make the right side equal to one.

$\frac{{(x + 1)}^{2}}{36} + \frac{9 {(y - 2)}^{2}}{36} = \frac{36}{36}$

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 $1$ .

$\frac{{(x + 1)}^{2}}{36} + \frac{{(y - 2)}^{2}}{4} = 1$

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.

$a = 6$

$b = 2$

$k = 2$

$h = - 1$

Step 4

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

$(- 1, 2)$

Step 5

Find the Center x^2+2x+9y^2-36y+1=0

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