Find the Center 48x^2+y^2+24y=0

$48 x^{2} + y^{2} + 24 y = 0$

Step 1

Find the standard form of the ellipse.

Complete the square for $y^{2} + 24 y$ .

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

$a = 1$

$b = 24$

$c = 0$

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

$d = \frac{24}{2 \cdot 1}$

Cancel the common factor of $24$ and $2$ .

Factor $2$ out of $24$ .

$d = \frac{2 \cdot 12}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot 12}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot 12}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{12}{1}$

Divide $12$ by $1$ .

$d = 12$

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

$e = 0 - \frac{24^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $24$ to the power of $2$ .

$e = 0 - \frac{576}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{576}{4}$

Divide $576$ by $4$ .

$e = 0 - 1 \cdot 144$

Multiply $- 1$ by $144$ .

$e = 0 - 144$

Subtract $144$ from $0$ .

$e = - 144$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y + 12)}^{2} - 144$ .

${(y + 12)}^{2} - 144$

Substitute ${(y + 12)}^{2} - 144$ for $y^{2} + 24 y$ in the equation $48 x^{2} + y^{2} + 24 y = 0$ .

$48 x^{2} + {(y + 12)}^{2} - 144 = 0$

Move $- 144$ to the right side of the equation by adding $144$ to both sides.

$48 x^{2} + {(y + 12)}^{2} = 0 + 144$

Add $0$ and $144$ .

$48 x^{2} + {(y + 12)}^{2} = 144$

Divide each term by $144$ to make the right side equal to one.

$\frac{48 x^{2}}{144} + \frac{{(y + 12)}^{2}}{144} = \frac{144}{144}$

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 $1$ .

$\frac{x^{2}}{3} + \frac{{(y + 12)}^{2}}{144} = 1$

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.

$a = 12$

$b = \sqrt{3}$

$k = - 12$

$h = 0$

Step 4

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

$(0, - 12)$

Step 5

Find the Center 48x^2+y^2+24y=0

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