Precalculus Examples

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

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 $1$ .

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

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.

$a = 2 \sqrt{3}$

$b = 2$

$k = 1$

$h = 1$

Step 4

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

$(1, 1)$

Step 5

Find $c$ , the distance from the center to a focus.

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Step 5.1

Find the distance from the center to a focus of the ellipse by using the following formula.

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

Step 5.2

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

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

Step 5.3

Simplify.

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Step 5.3.1

Simplify the expression.

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Step 5.3.1.1

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

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

Step 5.3.1.2

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

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

Step 5.3.2

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

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Step 5.3.2.1

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

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

Step 5.3.2.2

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

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

Step 5.3.2.3

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

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

Step 5.3.2.4

Cancel the common factor of $2$ .

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Step 5.3.2.4.1

Cancel the common factor.

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

Step 5.3.2.4.2

Rewrite the expression.

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

Step 5.3.2.5

Evaluate the exponent.

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

Step 5.3.3

Simplify the expression.

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Step 5.3.3.1

Multiply $4$ by $3$ .

$\sqrt{12 - {(2)}^{2}}$

Step 5.3.3.2

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

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

Step 5.3.3.3

Multiply $- 1$ by $4$ .

$\sqrt{12 - 4}$

Step 5.3.3.4

Subtract $4$ from $12$ .

$\sqrt{8}$

Step 5.3.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 5.3.4.1

Factor $4$ out of $8$ .

$\sqrt{4 (2)}$

Step 5.3.4.2

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

$\sqrt{2^{2} \cdot 2}$

Step 5.3.5

Pull terms out from under the radical.

$2 \sqrt{2}$

Step 6

Find the vertices.

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Step 6.1

The first vertex of an ellipse can be found by adding $a$ to $k$ .

$(h, k + a)$

Step 6.2

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(1, 1 + 2 \sqrt{3})$

Step 6.3

The second vertex of an ellipse can be found by subtracting $a$ from $k$ .

$(h, k - a)$

Step 6.4

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(1, 1 - (2 \sqrt{3}))$

Step 6.5

Simplify.

$(1, 1 - 2 \sqrt{3})$

Step 6.6

Ellipses have two vertices.

${Vertex}_{1}$ : $(1, 1 + 2 \sqrt{3})$

${Vertex}_{2}$ : $(1, 1 - 2 \sqrt{3})$

${Vertex}_{1}$ : $(1, 1 + 2 \sqrt{3})$

${Vertex}_{2}$ : $(1, 1 - 2 \sqrt{3})$

Step 7

Find the foci.

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Step 7.1

The first focus of an ellipse can be found by adding $c$ to $k$ .

$(h, k + c)$

Step 7.2

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(1, 1 + 2 \sqrt{2})$

Step 7.3

The first focus of an ellipse can be found by subtracting $c$ from $k$ .

$(h, k - c)$

Step 7.4

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(1, 1 - (2 \sqrt{2}))$

Step 7.5

Simplify.

$(1, 1 - 2 \sqrt{2})$

Step 7.6

Ellipses have two foci.

${Focus}_{1}$ : $(1, 1 + 2 \sqrt{2})$

${Focus}_{2}$ : $(1, 1 - 2 \sqrt{2})$

${Focus}_{1}$ : $(1, 1 + 2 \sqrt{2})$

${Focus}_{2}$ : $(1, 1 - 2 \sqrt{2})$

Step 8

Find the eccentricity.

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Step 8.1

Find the eccentricity by using the following formula.

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

Step 8.2

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

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

Step 8.3

Simplify.

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

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

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

Step 8.3.1.2

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

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

Step 8.3.1.3

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

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Step 8.3.1.3.1

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

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

Step 8.3.1.3.2

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

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

Step 8.3.1.3.3

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

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

Step 8.3.1.3.4

Cancel the common factor of $2$ .

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Step 8.3.1.3.4.1

Cancel the common factor.

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

Step 8.3.1.3.4.2

Rewrite the expression.

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

Step 8.3.1.3.5

Evaluate the exponent.

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

Step 8.3.1.4

Multiply $4$ by $3$ .

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

Step 8.3.1.5

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

$\frac{\sqrt{12 - 1 \cdot 4}}{2 \sqrt{3}}$

Step 8.3.1.6

Multiply $- 1$ by $4$ .

$\frac{\sqrt{12 - 4}}{2 \sqrt{3}}$

Step 8.3.1.7

Subtract $4$ from $12$ .

$\frac{\sqrt{8}}{2 \sqrt{3}}$

Step 8.3.1.8

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 8.3.1.8.1

Factor $4$ out of $8$ .

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

Step 8.3.1.8.2

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

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

Step 8.3.1.9

Pull terms out from under the radical.

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

Step 8.3.2

Cancel the common factor of $2$ .

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Step 8.3.2.1

Cancel the common factor.

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

Step 8.3.2.2

Rewrite the expression.

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

Step 8.3.3

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 8.3.4

Combine and simplify the denominator.

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Step 8.3.4.1

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 8.3.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 8.3.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 8.3.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 8.3.4.5

Add $1$ and $1$ .

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

Step 8.3.4.6

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

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Step 8.3.4.6.1

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

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

Step 8.3.4.6.2

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

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

Step 8.3.4.6.3

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

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

Step 8.3.4.6.4

Cancel the common factor of $2$ .

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Step 8.3.4.6.4.1

Cancel the common factor.

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

Step 8.3.4.6.4.2

Rewrite the expression.

$\frac{\sqrt{2} \sqrt{3}}{3^{1}}$

Step 8.3.4.6.5

Evaluate the exponent.

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

Step 8.3.5

Simplify the numerator.

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Step 8.3.5.1

Combine using the product rule for radicals.

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

Step 8.3.5.2

Multiply $2$ by $3$ .

$\frac{\sqrt{6}}{3}$

Step 9

These values represent the important values for graphing and analyzing an ellipse.

Center: $(1, 1)$

${Vertex}_{1}$ : $(1, 1 + 2 \sqrt{3})$

${Vertex}_{2}$ : $(1, 1 - 2 \sqrt{3})$

${Focus}_{1}$ : $(1, 1 + 2 \sqrt{2})$

${Focus}_{2}$ : $(1, 1 - 2 \sqrt{2})$

Eccentricity: $\frac{\sqrt{6}}{3}$

Step 10