Find the Eccentricity 9x^2+4y^2+36x-24y+36=0

$9 x^{2} + 4 y^{2} + 36 x - 24 y + 36 = 0$

Subtract $36$ from both sides of the equation.

$9 x^{2} + 4 y^{2} + 36 x - 24 y = - 36$

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

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 $9 {(x + 2)}^{2} - 36$ .

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

Substitute $9 {(x + 2)}^{2} - 36$ for $9 x^{2} + 36 x$ in the equation $9 x^{2} + 4 y^{2} + 36 x - 24 y = - 36$ .

$9 {(x + 2)}^{2} - 36 + 4 y^{2} - 24 y = - 36$

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

$9 {(x + 2)}^{2} + 4 y^{2} - 24 y = - 36 + 36$

Complete the square for $4 y^{2} - 24 y$ .

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

$a = 4$

$b = - 24$

$c = 0$

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

$d = \frac{- 24}{2 \cdot 4}$

Simplify the right side.

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

Factor $2$ out of $- 24$ .

$d = \frac{2 \cdot - 12}{2 \cdot 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot - 12}{2 (4)}$

Cancel the common factor.

$d = \frac{2 \cdot - 12}{2 \cdot 4}$

Rewrite the expression.

$d = \frac{- 12}{4}$

Cancel the common factor of $- 12$ and $4$ .

Factor $4$ out of $- 12$ .

$d = \frac{4 \cdot - 3}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$d = \frac{4 \cdot - 3}{4 (1)}$

Cancel the common factor.

$d = \frac{4 \cdot - 3}{4 \cdot 1}$

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

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

Simplify the right side.

Simplify each term.

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

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

Multiply $4$ by $4$ .

$e = 0 - \frac{576}{16}$

Divide $576$ by $16$ .

$e = 0 - 1 \cdot 36$

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 $4 {(y - 3)}^{2} - 36$ .

$4 {(y - 3)}^{2} - 36$

Substitute $4 {(y - 3)}^{2} - 36$ for $4 y^{2} - 24 y$ in the equation $9 x^{2} + 4 y^{2} + 36 x - 24 y = - 36$ .

$9 {(x + 2)}^{2} + 4 {(y - 3)}^{2} - 36 = - 36 + 36$

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

$9 {(x + 2)}^{2} + 4 {(y - 3)}^{2} = - 36 + 36 + 36$

Simplify $- 36 + 36 + 36$ .

Add $- 36$ and $36$ .

$9 {(x + 2)}^{2} + 4 {(y - 3)}^{2} = 0 + 36$

Add $0$ and $36$ .

$9 {(x + 2)}^{2} + 4 {(y - 3)}^{2} = 36$

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

$\frac{9 {(x + 2)}^{2}}{36} + \frac{4 {(y - 3)}^{2}}{36} = \frac{36}{36}$

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 $1$ .

$\frac{{(x + 2)}^{2}}{4} + \frac{{(y - 3)}^{2}}{9} = 1$

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.

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

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.

$a = 3$

$b = 2$

$k = 3$

$h = - 2$

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} - b^{2}}}{a}$

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(3)}^{2} - {(2)}^{2}}}{3}$

Simplify the numerator.

Raise $3$ to the power of $2$ .

$\frac{\sqrt{9 - 2^{2}}}{3}$

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

$\frac{\sqrt{9 - 1 \cdot 4}}{3}$

Multiply $- 1$ by $4$ .

$\frac{\sqrt{9 - 4}}{3}$

Subtract $4$ from $9$ .

$\frac{\sqrt{5}}{3}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{5}}{3}$

Decimal Form:

$0.74535599 \dots$

Find the Eccentricity 9x^2+4y^2+36x-24y+36=0

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