Algebra Examples

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

Step 1

Find the standard form of the ellipse.

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Step 1.1

Divide each term by

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

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

Step 1.2

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

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

$1$ .

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

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.

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

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.

$a = 2$

$b = \sqrt{3}$

$k = 0$

$h = 0$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

Raise

$2$ to the power of

$2$ .

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

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

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

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

Step 5.3.2.2

Apply the power rule and multiply exponents,

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

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

Step 5.3.2.3

Combine

$\frac{1}{2}$ and

$2$ .

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

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.

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

Step 5.3.2.4.2

Rewrite the expression.

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

Step 5.3.2.5

Evaluate the exponent.

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

Step 5.3.3

Simplify the expression.

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Step 5.3.3.1

Multiply

$- 1$ by

$3$ .

$\sqrt{4 - 3}$

Step 5.3.3.2

Subtract

$3$ from

$4$ .

$\sqrt{1}$

Step 5.3.3.3

Any root of

$1$ is

$1$ .

$1$

Step 6

Find the vertices.

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Step 6.1

The first vertex of an ellipse can be found by adding

$a$ to

$k$ .

$(h, k + a)$

Step 6.2

Substitute the known values of

$h$ ,

$a$ , and

$k$ into the formula.

$(0, 0 + 2)$

Step 6.3

Simplify.

$(0, 2)$

Step 6.4

The second vertex of an ellipse can be found by subtracting

$a$ from

$k$ .

$(h, k - a)$

Step 6.5

Substitute the known values of

$h$ ,

$a$ , and

$k$ into the formula.

$(0, 0 - (2))$

Step 6.6

Simplify.

$(0, - 2)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ :

$(0, 2)$

${Vertex}_{2}$ :

$(0, - 2)$

${Vertex}_{1}$ :

$(0, 2)$

${Vertex}_{2}$ :

$(0, - 2)$

Step 7

Find the foci.

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Step 7.1

The first focus of an ellipse can be found by adding

$c$ to

$k$ .

$(h, k + c)$

Step 7.2

Substitute the known values of

$h$ ,

$c$ , and

$k$ into the formula.

$(0, 0 + 1)$

Step 7.3

Simplify.

$(0, 1)$

Step 7.4

The first focus of an ellipse can be found by subtracting

$c$ from

$k$ .

$(h, k - c)$

Step 7.5

Substitute the known values of

$h$ ,

$c$ , and

$k$ into the formula.

$(0, 0 - (1))$

Step 7.6

Simplify.

$(0, - 1)$

Step 7.7

Ellipses have two foci.

${Focus}_{1}$ :

$(0, 1)$

${Focus}_{2}$ :

$(0, - 1)$

${Focus}_{1}$ :

$(0, 1)$

${Focus}_{2}$ :

$(0, - 1)$

Step 8

Find the eccentricity.

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Step 8.1

Find the eccentricity by using the following formula.

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

Step 8.2

Substitute the values of

$a$ and

$b$ into the formula.

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

Step 8.3

Simplify the numerator.

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Step 8.3.1

Raise

$2$ to the power of

$2$ .

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

Step 8.3.2

Rewrite

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

$3$ .

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Step 8.3.2.1

Use

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

$\sqrt{3}$ as

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

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

Step 8.3.2.2

Apply the power rule and multiply exponents,

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

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

Step 8.3.2.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 8.3.2.4

Cancel the common factor of

$2$ .

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Step 8.3.2.4.1

Cancel the common factor.

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

Step 8.3.2.4.2

Rewrite the expression.

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

Step 8.3.2.5

Evaluate the exponent.

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

Step 8.3.3

Multiply

$- 1$ by

$3$ .

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

Step 8.3.4

Subtract

$3$ from

$4$ .

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

Step 8.3.5

Any root of

$1$ is

$1$ .

$\frac{1}{2}$

Step 9

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

Center:

$(0, 0)$

${Vertex}_{1}$ :

$(0, 2)$

${Vertex}_{2}$ :

$(0, - 2)$

${Focus}_{1}$ :

$(0, 1)$

${Focus}_{2}$ :

$(0, - 1)$

Eccentricity:

$\frac{1}{2}$

Step 10

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