Pre-Algebra Examples

$4 x^{2} + 9 y^{2} = 81$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Divide each term by $81$ to make the right side equal to one.

$\frac{4 x^{2}}{81} + \frac{9 y^{2}}{81} = \frac{81}{81}$

Step 1.2

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 $1$ .

$\frac{x^{2}}{\frac{81}{4}} + \frac{y^{2}}{9} = 1$

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.

$a = \frac{9}{2}$

$b = 3$

$k = 0$

$h = 0$

Step 4

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

$(0, 0)$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

Apply the product rule to $\frac{9}{2}$ .

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

Step 5.3.2

Raise $9$ to the power of $2$ .

$\sqrt{\frac{81}{2^{2}} - {(3)}^{2}}$

Step 5.3.3

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

$\sqrt{\frac{81}{4} - {(3)}^{2}}$

Step 5.3.4

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

$\sqrt{\frac{81}{4} - 1 \cdot 9}$

Step 5.3.5

Multiply $- 1$ by $9$ .

$\sqrt{\frac{81}{4} - 9}$

Step 5.3.6

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{81}{4} - 9 \cdot \frac{4}{4}}$

Step 5.3.7

Combine $- 9$ and $\frac{4}{4}$ .

$\sqrt{\frac{81}{4} + \frac{- 9 \cdot 4}{4}}$

Step 5.3.8

Combine the numerators over the common denominator.

$\sqrt{\frac{81 - 9 \cdot 4}{4}}$

Step 5.3.9

Simplify the numerator.

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Step 5.3.9.1

Multiply $- 9$ by $4$ .

$\sqrt{\frac{81 - 36}{4}}$

Step 5.3.9.2

Subtract $36$ from $81$ .

$\sqrt{\frac{45}{4}}$

Step 5.3.10

Rewrite $\sqrt{\frac{45}{4}}$ as $\frac{\sqrt{45}}{\sqrt{4}}$ .

$\frac{\sqrt{45}}{\sqrt{4}}$

Step 5.3.11

Simplify the numerator.

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Step 5.3.11.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 5.3.11.1.1

Factor $9$ out of $45$ .

$\frac{\sqrt{9 (5)}}{\sqrt{4}}$

Step 5.3.11.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 5}}{\sqrt{4}}$

Step 5.3.11.2

Pull terms out from under the radical.

$\frac{3 \sqrt{5}}{\sqrt{4}}$

Step 5.3.12

Simplify the denominator.

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Step 5.3.12.1

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

$\frac{3 \sqrt{5}}{\sqrt{2^{2}}}$

Step 5.3.12.2

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

$\frac{3 \sqrt{5}}{2}$

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 $h$ .

$(h + a, k)$

Step 6.2

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

$(0 + \frac{9}{2}, 0)$

Step 6.3

Simplify.

$(\frac{9}{2}, 0)$

Step 6.4

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

$(h - a, k)$

Step 6.5

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

$(0 - (\frac{9}{2}), 0)$

Step 6.6

Simplify.

$(- \frac{9}{2}, 0)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(\frac{9}{2}, 0)$

${Vertex}_{2}$ : $(- \frac{9}{2}, 0)$

${Vertex}_{1}$ : $(\frac{9}{2}, 0)$

${Vertex}_{2}$ : $(- \frac{9}{2}, 0)$

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 $h$ .

$(h + c, k)$

Step 7.2

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

$(0 + \frac{3 \sqrt{5}}{2}, 0)$

Step 7.3

Simplify.

$(\frac{3 \sqrt{5}}{2}, 0)$

Step 7.4

The second focus of an ellipse can be found by subtracting $c$ from $h$ .

$(h - c, k)$

Step 7.5

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

$(0 - (\frac{3 \sqrt{5}}{2}), 0)$

Step 7.6

Simplify.

$(- \frac{3 \sqrt{5}}{2}, 0)$

Step 7.7

Ellipses have two foci.

${Focus}_{1}$ : $(\frac{3 \sqrt{5}}{2}, 0)$

${Focus}_{2}$ : $(- \frac{3 \sqrt{5}}{2}, 0)$

${Focus}_{1}$ : $(\frac{3 \sqrt{5}}{2}, 0)$

${Focus}_{2}$ : $(- \frac{3 \sqrt{5}}{2}, 0)$

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.

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

Step 8.3

Simplify.

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Step 8.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.3.2

Apply the product rule to $\frac{9}{2}$ .

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

Step 8.3.3

Raise $9$ to the power of $2$ .

$\sqrt{\frac{81}{2^{2}} - 3^{2}} \frac{2}{9}$

Step 8.3.4

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

$\sqrt{\frac{81}{4} - 3^{2}} \frac{2}{9}$

Step 8.3.5

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

$\sqrt{\frac{81}{4} - 1 \cdot 9} \frac{2}{9}$

Step 8.3.6

Multiply $- 1$ by $9$ .

$\sqrt{\frac{81}{4} - 9} \frac{2}{9}$

Step 8.3.7

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{81}{4} - 9 \cdot \frac{4}{4}} \frac{2}{9}$

Step 8.3.8

Combine $- 9$ and $\frac{4}{4}$ .

$\sqrt{\frac{81}{4} + \frac{- 9 \cdot 4}{4}} \frac{2}{9}$

Step 8.3.9

Combine the numerators over the common denominator.

$\sqrt{\frac{81 - 9 \cdot 4}{4}} \frac{2}{9}$

Step 8.3.10

Simplify the numerator.

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Step 8.3.10.1

Multiply $- 9$ by $4$ .

$\sqrt{\frac{81 - 36}{4}} \frac{2}{9}$

Step 8.3.10.2

Subtract $36$ from $81$ .

$\sqrt{\frac{45}{4}} \frac{2}{9}$

Step 8.3.11

Rewrite $\sqrt{\frac{45}{4}}$ as $\frac{\sqrt{45}}{\sqrt{4}}$ .

$\frac{\sqrt{45}}{\sqrt{4}} \cdot \frac{2}{9}$

Step 8.3.12

Simplify the numerator.

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Step 8.3.12.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 8.3.12.1.1

Factor $9$ out of $45$ .

$\frac{\sqrt{9 (5)}}{\sqrt{4}} \cdot \frac{2}{9}$

Step 8.3.12.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 5}}{\sqrt{4}} \cdot \frac{2}{9}$

Step 8.3.12.2

Pull terms out from under the radical.

$\frac{3 \sqrt{5}}{\sqrt{4}} \cdot \frac{2}{9}$

Step 8.3.13

Simplify the denominator.

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Step 8.3.13.1

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

$\frac{3 \sqrt{5}}{\sqrt{2^{2}}} \cdot \frac{2}{9}$

Step 8.3.13.2

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

$\frac{3 \sqrt{5}}{2} \cdot \frac{2}{9}$

Step 8.3.14

Simplify terms.

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Step 8.3.14.1

Cancel the common factor of $3$ .

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Step 8.3.14.1.1

Factor $3$ out of $3 \sqrt{5}$ .

$\frac{3 (\sqrt{5})}{2} \cdot \frac{2}{9}$

Step 8.3.14.1.2

Factor $3$ out of $9$ .

$\frac{3 (\sqrt{5})}{2} \cdot \frac{2}{3 (3)}$

Step 8.3.14.1.3

Cancel the common factor.

$\frac{3 \sqrt{5}}{2} \cdot \frac{2}{3 \cdot 3}$

Step 8.3.14.1.4

Rewrite the expression.

$\frac{\sqrt{5}}{2} \cdot \frac{2}{3}$

Step 8.3.14.2

Cancel the common factor of $2$ .

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Step 8.3.14.2.1

Cancel the common factor.

$\frac{\sqrt{5}}{2} \cdot \frac{2}{3}$

Step 8.3.14.2.2

Rewrite the expression.

$\sqrt{5} \frac{1}{3}$

Step 8.3.14.3

Combine $\sqrt{5}$ and $\frac{1}{3}$ .

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

Step 9

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

Center: $(0, 0)$

${Vertex}_{1}$ : $(\frac{9}{2}, 0)$

${Vertex}_{2}$ : $(- \frac{9}{2}, 0)$

${Focus}_{1}$ : $(\frac{3 \sqrt{5}}{2}, 0)$

${Focus}_{2}$ : $(- \frac{3 \sqrt{5}}{2}, 0)$

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

Step 10