Find the Eccentricity ((x+3)^2)/12+((y-2)^2)/16=1

$\frac{{(x + 3)}^{2}}{12} + \frac{{(y - 2)}^{2}}{16} = 1$

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 $1$ .

$\frac{{(x + 3)}^{2}}{12} + \frac{{(y - 2)}^{2}}{16} = 1$

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.

$a = 4$

$b = 2 \sqrt{3}$

$k = 2$

$h = - 3$

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.

$\frac{\sqrt{{(4)}^{2} - {(2 \sqrt{3})}^{2}}}{4}$

Simplify.

Simplify the numerator.

Raise $4$ to the power of $2$ .

$\frac{\sqrt{16 - {(2 \sqrt{3})}^{2}}}{4}$

Apply the product rule to $2 \sqrt{3}$ .

$\frac{\sqrt{16 - (2^{2} {\sqrt{3}}^{2})}}{4}$

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

$\frac{\sqrt{16 - (4 {\sqrt{3}}^{2})}}{4}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{16 - (4 {(3^{\frac{1}{2}})}^{2})}}{4}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{16 - (4 \cdot 3^{\frac{1}{2} \cdot 2})}}{4}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{16 - (4 \cdot 3^{\frac{2}{2}})}}{4}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{16 - (4 \cdot 3^{\frac{2}{2}})}}{4}$

Rewrite the expression.

$\frac{\sqrt{16 - (4 \cdot 3^{1})}}{4}$

Evaluate the exponent.

$\frac{\sqrt{16 - (4 \cdot 3)}}{4}$

Multiply $4$ by $3$ .

$\frac{\sqrt{16 - 1 \cdot 12}}{4}$

Multiply $- 1$ by $12$ .

$\frac{\sqrt{16 - 12}}{4}$

Subtract $12$ from $16$ .

$\frac{\sqrt{4}}{4}$

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2}}}{4}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{2}{4}$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$\frac{2 (1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{2}$

Find the Eccentricity ((x+3)^2)/12+((y-2)^2)/16=1

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