Find the Center 2x^2+3y^2-20x+18y+71=0

$2 x^{2} + 3 y^{2} - 20 x + 18 y + 71 = 0$

Step 1

Find the standard form of the ellipse.

Subtract $71$ from both sides of the equation.

$2 x^{2} + 3 y^{2} - 20 x + 18 y = - 71$

Complete the square for $2 x^{2} - 20 x$ .

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

$a = 2$

$b = - 20$

$c = 0$

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

$d = \frac{- 20}{2 \cdot 2}$

Simplify the right side.

Cancel the common factor of $- 20$ and $2$ .

Factor $2$ out of $- 20$ .

$d = \frac{2 \cdot - 10}{2 \cdot 2}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 10}{2 (2)}$

Cancel the common factor.

$d = \frac{2 \cdot - 10}{2 \cdot 2}$

Rewrite the expression.

$d = \frac{- 10}{2}$

Cancel the common factor of $- 10$ and $2$ .

Factor $2$ out of $- 10$ .

$d = \frac{2 \cdot - 5}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$d = \frac{2 \cdot - 5}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

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

$e = 0 - \frac{{(- 20)}^{2}}{4 \cdot 2}$

Simplify the right side.

Simplify each term.

Raise $- 20$ to the power of $2$ .

$e = 0 - \frac{400}{4 \cdot 2}$

Multiply $4$ by $2$ .

$e = 0 - \frac{400}{8}$

Divide $400$ by $8$ .

$e = 0 - 1 \cdot 50$

Multiply $- 1$ by $50$ .

$e = 0 - 50$

Subtract $50$ from $0$ .

$e = - 50$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x - 5)}^{2} - 50$ .

$2 {(x - 5)}^{2} - 50$

Substitute $2 {(x - 5)}^{2} - 50$ for $2 x^{2} - 20 x$ in the equation $2 x^{2} + 3 y^{2} - 20 x + 18 y = - 71$ .

$2 {(x - 5)}^{2} - 50 + 3 y^{2} + 18 y = - 71$

Move $- 50$ to the right side of the equation by adding $50$ to both sides.

$2 {(x - 5)}^{2} + 3 y^{2} + 18 y = - 71 + 50$

Complete the square for $3 y^{2} + 18 y$ .

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

$a = 3$

$b = 18$

$c = 0$

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

$d = \frac{18}{2 \cdot 3}$

Simplify the right side.

Cancel the common factor of $18$ and $2$ .

Factor $2$ out of $18$ .

$d = \frac{2 \cdot 9}{2 \cdot 3}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 3$ .

$d = \frac{2 \cdot 9}{2 (3)}$

Cancel the common factor.

$d = \frac{2 \cdot 9}{2 \cdot 3}$

Rewrite the expression.

$d = \frac{9}{3}$

Cancel the common factor of $9$ and $3$ .

Factor $3$ out of $9$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

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

$e = 0 - \frac{18^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $18$ to the power of $2$ .

$e = 0 - \frac{324}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 0 - \frac{324}{12}$

Divide $324$ by $12$ .

$e = 0 - 1 \cdot 27$

Multiply $- 1$ by $27$ .

$e = 0 - 27$

Subtract $27$ from $0$ .

$e = - 27$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(y + 3)}^{2} - 27$ .

$3 {(y + 3)}^{2} - 27$

Substitute $3 {(y + 3)}^{2} - 27$ for $3 y^{2} + 18 y$ in the equation $2 x^{2} + 3 y^{2} - 20 x + 18 y = - 71$ .

$2 {(x - 5)}^{2} + 3 {(y + 3)}^{2} - 27 = - 71 + 50$

Move $- 27$ to the right side of the equation by adding $27$ to both sides.

$2 {(x - 5)}^{2} + 3 {(y + 3)}^{2} = - 71 + 50 + 27$

Simplify $- 71 + 50 + 27$ .

Add $- 71$ and $50$ .

$2 {(x - 5)}^{2} + 3 {(y + 3)}^{2} = - 21 + 27$

Add $- 21$ and $27$ .

$2 {(x - 5)}^{2} + 3 {(y + 3)}^{2} = 6$

Divide each term by $6$ to make the right side equal to one.

$\frac{2 {(x - 5)}^{2}}{6} + \frac{3 {(y + 3)}^{2}}{6} = \frac{6}{6}$

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 $1$ .

$\frac{{(x - 5)}^{2}}{3} + \frac{{(y + 3)}^{2}}{2} = 1$

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.

$a = \sqrt{3}$

$b = \sqrt{2}$

$k = - 3$

$h = 5$

Step 4

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

$(5, - 3)$

Step 5

Find the Center 2x^2+3y^2-20x+18y+71=0

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