Precalculus Examples

$\frac{{(x + 2)}^{2}}{12} + \frac{{(y - 4)}^{2}}{9} = 1$

Step 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 $1$ .

$\frac{{(x + 2)}^{2}}{12} + \frac{{(y - 4)}^{2}}{9} = 1$

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.

$a = 2 \sqrt{3}$

$b = 3$

$k = 4$

$h = - 2$

Step 4

Find the eccentricity by using the following formula.

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

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

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Step 6.1.3.1

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

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

Step 6.1.3.2

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

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

Step 6.1.3.3

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

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

Step 6.1.3.4

Cancel the common factor of $2$ .

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Step 6.1.3.4.1

Cancel the common factor.

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

Step 6.1.3.4.2

Rewrite the expression.

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

Step 6.1.3.5

Evaluate the exponent.

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

Step 6.1.4

Multiply $4$ by $3$ .

$\frac{\sqrt{12 - 3^{2}}}{2 \sqrt{3}}$

Step 6.1.5

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

$\frac{\sqrt{12 - 1 \cdot 9}}{2 \sqrt{3}}$

Step 6.1.6

Multiply $- 1$ by $9$ .

$\frac{\sqrt{12 - 9}}{2 \sqrt{3}}$

Step 6.1.7

Subtract $9$ from $12$ .

$\frac{\sqrt{3}}{2 \sqrt{3}}$

Step 6.2

Cancel the common factor of $\sqrt{3}$ .

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Step 6.2.1

Cancel the common factor.

$\frac{\sqrt{3}}{2 \sqrt{3}}$

Step 6.2.2

Rewrite the expression.

$\frac{1}{2}$

Step 7