Basic Math Examples

$\frac{x^{2}}{2} + \frac{{(y - 8)}^{2}}{64} = 1$

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 $1$ .

$\frac{x^{2}}{2} + \frac{{(y - 8)}^{2}}{64} = 1$

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.

$a = 8$

$b = \sqrt{2}$

$k = 8$

$h = 0$

Step 4

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

$(0, 8)$

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.

$\sqrt{{(8)}^{2} - {(\sqrt{2})}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Raise $8$ to the power of $2$ .

$\sqrt{64 - {(\sqrt{2})}^{2}}$

Step 5.3.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 5.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sqrt{64 - {(2^{\frac{1}{2}})}^{2}}$

Step 5.3.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{64 - 2^{\frac{1}{2} \cdot 2}}$

Step 5.3.2.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{64 - 2^{\frac{2}{2}}}$

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.

$\sqrt{64 - 2^{\frac{2}{2}}}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{64 - 2^{1}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{64 - 1 \cdot 2}$

Step 5.3.3

Simplify the expression.

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Step 5.3.3.1

Multiply $- 1$ by $2$ .

$\sqrt{64 - 2}$

Step 5.3.3.2

Subtract $2$ from $64$ .

$\sqrt{62}$

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.

$(0, 8 + 8)$

Step 6.3

Simplify.

$(0, 16)$

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.

$(0, 8 - (8))$

Step 6.6

Simplify.

$(0, 0)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(0, 16)$

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

${Vertex}_{1}$ : $(0, 16)$

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

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 $k$ .

$(h, k + c)$

Step 7.2

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

$(0, 8 + \sqrt{62})$

Step 7.3

The first focus of an ellipse can be found by subtracting $c$ from $k$ .

$(h, k - c)$

Step 7.4

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

$(0, 8 - (\sqrt{62}))$

Step 7.5

Simplify.

$(0, 8 - \sqrt{62})$

Step 7.6

Ellipses have two foci.

${Focus}_{1}$ : $(0, 8 + \sqrt{62})$

${Focus}_{2}$ : $(0, 8 - \sqrt{62})$

${Focus}_{1}$ : $(0, 8 + \sqrt{62})$

${Focus}_{2}$ : $(0, 8 - \sqrt{62})$

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.

$\frac{\sqrt{{(8)}^{2} - {(\sqrt{2})}^{2}}}{8}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Raise $8$ to the power of $2$ .

$\frac{\sqrt{64 - {\sqrt{2}}^{2}}}{8}$

Step 8.3.2

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 8.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\sqrt{64 - {(2^{\frac{1}{2}})}^{2}}}{8}$

Step 8.3.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{64 - 2^{\frac{1}{2} \cdot 2}}}{8}$

Step 8.3.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{64 - 2^{\frac{2}{2}}}}{8}$

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.

$\frac{\sqrt{64 - 2^{\frac{2}{2}}}}{8}$

Step 8.3.2.4.2

Rewrite the expression.

$\frac{\sqrt{64 - 2^{1}}}{8}$

Step 8.3.2.5

Evaluate the exponent.

$\frac{\sqrt{64 - 1 \cdot 2}}{8}$

Step 8.3.3

Multiply $- 1$ by $2$ .

$\frac{\sqrt{64 - 2}}{8}$

Step 8.3.4

Subtract $2$ from $64$ .

$\frac{\sqrt{62}}{8}$

Step 9

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

Center: $(0, 8)$

${Vertex}_{1}$ : $(0, 16)$

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

${Focus}_{1}$ : $(0, 8 + \sqrt{62})$

${Focus}_{2}$ : $(0, 8 - \sqrt{62})$

Eccentricity: $\frac{\sqrt{62}}{8}$

Step 10