Precalculus Examples

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

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 $1$ .

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

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.

$a = 9$

$b = 3$

$k = 0$

$h = 0$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

Multiply $- 1$ by $9$ .

$\sqrt{81 - 9}$

Step 5.3.4

Subtract $9$ from $81$ .

$\sqrt{72}$

Step 5.3.5

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 5.3.5.1

Factor $36$ out of $72$ .

$\sqrt{36 (2)}$

Step 5.3.5.2

Rewrite $36$ as $6^{2}$ .

$\sqrt{6^{2} \cdot 2}$

Step 5.3.6

Pull terms out from under the radical.

$6 \sqrt{2}$

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 $h$ .

$(h + a, k)$

Step 6.2

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

$(0 + 9, 0)$

Step 6.3

Simplify.

$(9, 0)$

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.

$(0 - (9), 0)$

Step 6.6

Simplify.

$(- 9, 0)$

Step 6.7

Ellipses have two vertices.

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

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

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

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

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.

$(0 + 6 \sqrt{2}, 0)$

Step 7.3

Simplify.

$(6 \sqrt{2}, 0)$

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.

$(0 - (6 \sqrt{2}), 0)$

Step 7.6

Simplify.

$(- 6 \sqrt{2}, 0)$

Step 7.7

Ellipses have two foci.

${Focus}_{1}$ : $(6 \sqrt{2}, 0)$

${Focus}_{2}$ : $(- 6 \sqrt{2}, 0)$

${Focus}_{1}$ : $(6 \sqrt{2}, 0)$

${Focus}_{2}$ : $(- 6 \sqrt{2}, 0)$

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.

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

Step 8.3

Simplify.

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

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

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

Step 8.3.1.2

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

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

Step 8.3.1.3

Multiply $- 1$ by $9$ .

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

Step 8.3.1.4

Subtract $9$ from $81$ .

$\frac{\sqrt{72}}{9}$

Step 8.3.1.5

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 8.3.1.5.1

Factor $36$ out of $72$ .

$\frac{\sqrt{36 (2)}}{9}$

Step 8.3.1.5.2

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{6^{2} \cdot 2}}{9}$

Step 8.3.1.6

Pull terms out from under the radical.

$\frac{6 \sqrt{2}}{9}$

Step 8.3.2

Cancel the common factor of $6$ and $9$ .

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Step 8.3.2.1

Factor $3$ out of $6 \sqrt{2}$ .

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

Step 8.3.2.2

Cancel the common factors.

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Step 8.3.2.2.1

Factor $3$ out of $9$ .

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

Step 8.3.2.2.2

Cancel the common factor.

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

Step 8.3.2.2.3

Rewrite the expression.

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

Step 9

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

Center: $(0, 0)$

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

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

${Focus}_{1}$ : $(6 \sqrt{2}, 0)$

${Focus}_{2}$ : $(- 6 \sqrt{2}, 0)$

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

Step 10