Examples

$x^{2} + 4 y^{2} = 16$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Divide each term by $16$ to make the right side equal to one.

$\frac{x^{2}}{16} + \frac{4 y^{2}}{16} = \frac{16}{16}$

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 $1$ .

$\frac{x^{2}}{16} + \frac{y^{2}}{4} = 1$

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.

$a = 4$

$b = 2$

$k = 0$

$h = 0$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

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

$\sqrt{16 - {(2)}^{2}}$

Step 5.3.2

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

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

Step 5.3.3

Multiply $- 1$ by $4$ .

$\sqrt{16 - 4}$

Step 5.3.4

Subtract $4$ from $16$ .

$\sqrt{12}$

Step 5.3.5

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.3.5.1

Factor $4$ out of $12$ .

$\sqrt{4 (3)}$

Step 5.3.5.2

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

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

Step 5.3.6

Pull terms out from under the radical.

$2 \sqrt{3}$

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.

$(0 + 4, 0)$

Step 6.3

Simplify.

$(4, 0)$

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.

$(0 - (4), 0)$

Step 6.6

Simplify.

$(- 4, 0)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(4, 0)$

${Vertex}_{2}$ : $(- 4, 0)$

${Vertex}_{1}$ : $(4, 0)$

${Vertex}_{2}$ : $(- 4, 0)$

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.

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

Step 7.3

Simplify.

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

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.

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

Step 7.6

Simplify.

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

Step 7.7

Ellipses have two foci.

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

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

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

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

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.

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

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 $4$ to the power of $2$ .

$\frac{\sqrt{16 - 2^{2}}}{4}$

Step 8.3.1.2

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

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

Step 8.3.1.3

Multiply $- 1$ by $4$ .

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

Step 8.3.1.4

Subtract $4$ from $16$ .

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

Step 8.3.1.5

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 8.3.1.5.1

Factor $4$ out of $12$ .

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

Step 8.3.1.5.2

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

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

Step 8.3.1.6

Pull terms out from under the radical.

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

Step 8.3.2

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

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Step 8.3.2.1

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

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

Step 8.3.2.2

Cancel the common factors.

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Step 8.3.2.2.1

Factor $2$ out of $4$ .

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

Step 8.3.2.2.2

Cancel the common factor.

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

Step 8.3.2.2.3

Rewrite the expression.

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

Step 9

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

Center: $(0, 0)$

${Vertex}_{1}$ : $(4, 0)$

${Vertex}_{2}$ : $(- 4, 0)$

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

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

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

Step 10

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