Examples

4 x^{2} + 3 y^{2} = 12

$4 x^{2} + 3 y^{2} = 12$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Divide each term by

12

$12$ to make the right side equal to one.

\frac{4 x^{2}}{12} + \frac{3 y^{2}}{12} = \frac{12}{12}

$\frac{4 x^{2}}{12} + \frac{3 y^{2}}{12} = \frac{12}{12}$

Step 1.2

Simplify each term in the equation in order to set the right side equal to

1

$1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be

1

$1$ .

\frac{x^{2}}{3} + \frac{y^{2}}{4} = 1

$\frac{x^{2}}{3} + \frac{y^{2}}{4} = 1$

\frac{x^{2}}{3} + \frac{y^{2}}{4} = 1

$\frac{x^{2}}{3} + \frac{y^{2}}{4} = 1$

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}}{b^{2}} + \frac{{(y - k)}^{2}}{a^{2}} = 1

$\frac{{(x - h)}^{2}}{b^{2}} + \frac{{(y - k)}^{2}}{a^{2}} = 1$

Step 3

Match the values in this ellipse to those of the standard form. The variable

a

$a$ represents the radius of the major axis of the ellipse,

b

$b$ represents the radius of the minor axis of the ellipse,

h

$h$ represents the x-offset from the origin, and

k

$k$ represents the y-offset from the origin.

a = 2

$a = 2$

b = \sqrt{3}

$b = \sqrt{3}$

k = 0

$k = 0$

h = 0

$h = 0$

Step 4

The center of an ellipse follows the form of

(h, k)

$(h, k)$ . Substitute in the values of

h

$h$ and

k

$k$ .

(0, 0)

$(0, 0)$

Step 5

Find

c

$c$ , 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}}

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

Step 5.2

Substitute the values of

a

$a$ and

b

$b$ in the formula.

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

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

Step 5.3

Simplify.

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Step 5.3.1

Raise

2

$2$ to the power of

2

$2$ .

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

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

Step 5.3.2

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 5.3.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{3}

$\sqrt{3}$ as

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

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

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

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

Step 5.3.2.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 5.3.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 5.3.2.4

Cancel the common factor of

2

$2$ .

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Step 5.3.2.4.1

Cancel the common factor.

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

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

Step 5.3.2.4.2

Rewrite the expression.

\sqrt{4 - 3^{1}}

$\sqrt{4 - 3^{1}}$

\sqrt{4 - 3^{1}}

$\sqrt{4 - 3^{1}}$

Step 5.3.2.5

Evaluate the exponent.

\sqrt{4 - 1 \cdot 3}

$\sqrt{4 - 1 \cdot 3}$

\sqrt{4 - 1 \cdot 3}

$\sqrt{4 - 1 \cdot 3}$

Step 5.3.3

Simplify the expression.

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Step 5.3.3.1

Multiply

- 1

$- 1$ by

3

$3$ .

\sqrt{4 - 3}

$\sqrt{4 - 3}$

Step 5.3.3.2

Subtract

3

$3$ from

4

$4$ .

\sqrt{1}

$\sqrt{1}$

Step 5.3.3.3

Any root of

1

$1$ is

1

$1$ .

1

$1$

1

$1$

1

$1$

1

$1$

Step 6

Find the vertices.

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Step 6.1

The first vertex of an ellipse can be found by adding

a

$a$ to

k

$k$ .

(h, k + a)

$(h, k + a)$

Step 6.2

Substitute the known values of

h

$h$ ,

a

$a$ , and

k

$k$ into the formula.

(0, 0 + 2)

$(0, 0 + 2)$

Step 6.3

Simplify.

(0, 2)

$(0, 2)$

Step 6.4

The second vertex of an ellipse can be found by subtracting

a

$a$ from

k

$k$ .

(h, k - a)

$(h, k - a)$

Step 6.5

Substitute the known values of

h

$h$ ,

a

$a$ , and

k

$k$ into the formula.

(0, 0 - (2))

$(0, 0 - (2))$

Step 6.6

Simplify.

(0, - 2)

$(0, - 2)$

Step 6.7

Ellipses have two vertices.

{Vertex}_{1}

${Vertex}_{1}$ :

(0, 2)

$(0, 2)$

{Vertex}_{2}

${Vertex}_{2}$ :

(0, - 2)

$(0, - 2)$

{Vertex}_{1}

${Vertex}_{1}$ :

(0, 2)

$(0, 2)$

{Vertex}_{2}

${Vertex}_{2}$ :

(0, - 2)

$(0, - 2)$

Step 7

Find the foci.

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Step 7.1

The first focus of an ellipse can be found by adding

c

$c$ to

k

$k$ .

(h, k + c)

$(h, k + c)$

Step 7.2

Substitute the known values of

h

$h$ ,

c

$c$ , and

k

$k$ into the formula.

(0, 0 + 1)

$(0, 0 + 1)$

Step 7.3

Simplify.

(0, 1)

$(0, 1)$

Step 7.4

The first focus of an ellipse can be found by subtracting

c

$c$ from

k

$k$ .

(h, k - c)

$(h, k - c)$

Step 7.5

Substitute the known values of

h

$h$ ,

c

$c$ , and

k

$k$ into the formula.

(0, 0 - (1))

$(0, 0 - (1))$

Step 7.6

Simplify.

(0, - 1)

$(0, - 1)$

Step 7.7

Ellipses have two foci.

{Focus}_{1}

${Focus}_{1}$ :

(0, 1)

$(0, 1)$

{Focus}_{2}

${Focus}_{2}$ :

(0, - 1)

$(0, - 1)$

{Focus}_{1}

${Focus}_{1}$ :

(0, 1)

$(0, 1)$

{Focus}_{2}

${Focus}_{2}$ :

(0, - 1)

$(0, - 1)$

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}

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

Step 8.2

Substitute the values of

a

$a$ and

b

$b$ into the formula.

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

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

Step 8.3

Simplify the numerator.

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Step 8.3.1

Raise

2

$2$ to the power of

2

$2$ .

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

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

Step 8.3.2

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 8.3.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{3}

$\sqrt{3}$ as

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

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

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

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

Step 8.3.2.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 8.3.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 8.3.2.4

Cancel the common factor of

2

$2$ .

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Step 8.3.2.4.1

Cancel the common factor.

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

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

Step 8.3.2.4.2

Rewrite the expression.

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

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

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

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

Step 8.3.2.5

Evaluate the exponent.

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

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

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

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

Step 8.3.3

Multiply

- 1

$- 1$ by

3

$3$ .

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

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

Step 8.3.4

Subtract

3

$3$ from

4

$4$ .

\frac{\sqrt{1}}{2}

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

Step 8.3.5

Any root of

1

$1$ is

1

$1$ .

\frac{1}{2}

$\frac{1}{2}$

\frac{1}{2}

$\frac{1}{2}$

\frac{1}{2}

$\frac{1}{2}$

Step 9

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

Center:

(0, 0)

$(0, 0)$

{Vertex}_{1}

${Vertex}_{1}$ :

(0, 2)

$(0, 2)$

{Vertex}_{2}

${Vertex}_{2}$ :

(0, - 2)

$(0, - 2)$

{Focus}_{1}

${Focus}_{1}$ :

(0, 1)

$(0, 1)$

{Focus}_{2}

${Focus}_{2}$ :

(0, - 1)

$(0, - 1)$

Eccentricity:

\frac{1}{2}

$\frac{1}{2}$

Step 10

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