Precalculus Examples

$2 x^{2} + 7 y^{2} = 14$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Divide each term by $14$ to make the right side equal to one.

$\frac{2 x^{2}}{14} + \frac{7 y^{2}}{14} = \frac{14}{14}$

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 $1$ .

$\frac{x^{2}}{7} + \frac{y^{2}}{2} = 1$

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.

$a = \sqrt{7}$

$b = \sqrt{2}$

$k = 0$

$h = 0$

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.

$\sqrt{{(\sqrt{7})}^{2} - {(\sqrt{2})}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 5.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{{(7^{\frac{1}{2}})}^{2} - {(\sqrt{2})}^{2}}$

Step 5.3.1.2

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

$\sqrt{7^{\frac{1}{2} \cdot 2} - {(\sqrt{2})}^{2}}$

Step 5.3.1.3

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

$\sqrt{7^{\frac{2}{2}} - {(\sqrt{2})}^{2}}$

Step 5.3.1.4

Cancel the common factor of $2$ .

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Step 5.3.1.4.1

Cancel the common factor.

$\sqrt{7^{\frac{2}{2}} - {(\sqrt{2})}^{2}}$

Step 5.3.1.4.2

Rewrite the expression.

$\sqrt{7^{1} - {(\sqrt{2})}^{2}}$

Step 5.3.1.5

Evaluate the exponent.

$\sqrt{7 - {(\sqrt{2})}^{2}}$

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

$\sqrt{7 - {(2^{\frac{1}{2}})}^{2}}$

Step 5.3.2.2

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

$\sqrt{7 - 2^{\frac{1}{2} \cdot 2}}$

Step 5.3.2.3

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

$\sqrt{7 - 2^{\frac{2}{2}}}$

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.

$\sqrt{7 - 2^{\frac{2}{2}}}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{7 - 2^{1}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{7 - 1 \cdot 2}$

Step 5.3.3

Simplify the expression.

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Step 5.3.3.1

Multiply $- 1$ by $2$ .

$\sqrt{7 - 2}$

Step 5.3.3.2

Subtract $2$ from $7$ .

$\sqrt{5}$

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.

$(0 + \sqrt{7}, 0)$

Step 6.3

Simplify.

$(\sqrt{7}, 0)$

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.

$(0 - (\sqrt{7}), 0)$

Step 6.6

Simplify.

$(- \sqrt{7}, 0)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(\sqrt{7}, 0)$

${Vertex}_{2}$ : $(- \sqrt{7}, 0)$

${Vertex}_{1}$ : $(\sqrt{7}, 0)$

${Vertex}_{2}$ : $(- \sqrt{7}, 0)$

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.

$(0 + \sqrt{5}, 0)$

Step 7.3

Simplify.

$(\sqrt{5}, 0)$

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.

$(0 - (\sqrt{5}), 0)$

Step 7.6

Simplify.

$(- \sqrt{5}, 0)$

Step 7.7

Ellipses have two foci.

${Focus}_{1}$ : $(\sqrt{5}, 0)$

${Focus}_{2}$ : $(- \sqrt{5}, 0)$

${Focus}_{1}$ : $(\sqrt{5}, 0)$

${Focus}_{2}$ : $(- \sqrt{5}, 0)$

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.

$\frac{\sqrt{{(\sqrt{7})}^{2} - {(\sqrt{2})}^{2}}}{\sqrt{7}}$

Step 8.3

Simplify.

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 8.3.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{\sqrt{{(7^{\frac{1}{2}})}^{2} - {\sqrt{2}}^{2}}}{\sqrt{7}}$

Step 8.3.1.1.2

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

$\frac{\sqrt{7^{\frac{1}{2} \cdot 2} - {\sqrt{2}}^{2}}}{\sqrt{7}}$

Step 8.3.1.1.3

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

$\frac{\sqrt{7^{\frac{2}{2}} - {\sqrt{2}}^{2}}}{\sqrt{7}}$

Step 8.3.1.1.4

Cancel the common factor of $2$ .

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Step 8.3.1.1.4.1

Cancel the common factor.

$\frac{\sqrt{7^{\frac{2}{2}} - {\sqrt{2}}^{2}}}{\sqrt{7}}$

Step 8.3.1.1.4.2

Rewrite the expression.

$\frac{\sqrt{7^{1} - {\sqrt{2}}^{2}}}{\sqrt{7}}$

Step 8.3.1.1.5

Evaluate the exponent.

$\frac{\sqrt{7 - {\sqrt{2}}^{2}}}{\sqrt{7}}$

Step 8.3.1.2

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

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Step 8.3.1.2.1

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

$\frac{\sqrt{7 - {(2^{\frac{1}{2}})}^{2}}}{\sqrt{7}}$

Step 8.3.1.2.2

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

$\frac{\sqrt{7 - 2^{\frac{1}{2} \cdot 2}}}{\sqrt{7}}$

Step 8.3.1.2.3

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

$\frac{\sqrt{7 - 2^{\frac{2}{2}}}}{\sqrt{7}}$

Step 8.3.1.2.4

Cancel the common factor of $2$ .

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Step 8.3.1.2.4.1

Cancel the common factor.

$\frac{\sqrt{7 - 2^{\frac{2}{2}}}}{\sqrt{7}}$

Step 8.3.1.2.4.2

Rewrite the expression.

$\frac{\sqrt{7 - 2^{1}}}{\sqrt{7}}$

Step 8.3.1.2.5

Evaluate the exponent.

$\frac{\sqrt{7 - 1 \cdot 2}}{\sqrt{7}}$

Step 8.3.1.3

Multiply $- 1$ by $2$ .

$\frac{\sqrt{7 - 2}}{\sqrt{7}}$

Step 8.3.1.4

Subtract $2$ from $7$ .

$\frac{\sqrt{5}}{\sqrt{7}}$

Step 8.3.2

Multiply $\frac{\sqrt{5}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\sqrt{5}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 8.3.3

Combine and simplify the denominator.

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Step 8.3.3.1

Multiply $\frac{\sqrt{5}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\sqrt{5} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 8.3.3.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\sqrt{5} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 8.3.3.3

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\sqrt{5} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 8.3.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{5} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 8.3.3.5

Add $1$ and $1$ .

$\frac{\sqrt{5} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 8.3.3.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 8.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{\sqrt{5} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 8.3.3.6.2

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

$\frac{\sqrt{5} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 8.3.3.6.3

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

$\frac{\sqrt{5} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 8.3.3.6.4

Cancel the common factor of $2$ .

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Step 8.3.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{5} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 8.3.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{5} \sqrt{7}}{7^{1}}$

Step 8.3.3.6.5

Evaluate the exponent.

$\frac{\sqrt{5} \sqrt{7}}{7}$

Step 8.3.4

Simplify the numerator.

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Step 8.3.4.1

Combine using the product rule for radicals.

$\frac{\sqrt{5 \cdot 7}}{7}$

Step 8.3.4.2

Multiply $5$ by $7$ .

$\frac{\sqrt{35}}{7}$

Step 9

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

Center: $(0, 0)$

${Vertex}_{1}$ : $(\sqrt{7}, 0)$

${Vertex}_{2}$ : $(- \sqrt{7}, 0)$

${Focus}_{1}$ : $(\sqrt{5}, 0)$

${Focus}_{2}$ : $(- \sqrt{5}, 0)$

Eccentricity: $\frac{\sqrt{35}}{7}$

Step 10