Pre-Algebra Examples

$\frac{{(x - 1)}^{2}}{27} - \frac{{(y + 2)}^{2}}{169} = 1$

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 $1$ .

$\frac{{(x - 1)}^{2}}{27} - \frac{{(y + 2)}^{2}}{169} = 1$

Step 2

This is the form of a hyperbola. Use this form to determine the values used to find vertices and asymptotes of the hyperbola.

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

Step 3

Match the values in this hyperbola to those of the standard form. The variable $h$ represents the x-offset from the origin, $k$ represents the y-offset from origin, $a$ .

$a = 3 \sqrt{3}$

$b = 13$

$k = - 2$

$h = 1$

Step 4

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

$(1, - 2)$

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 hyperbola by using the following formula.

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

Step 5.2

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(3 \sqrt{3})}^{2} + {(13)}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the expression.

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Step 5.3.1.1

Apply the product rule to $3 \sqrt{3}$ .

$\sqrt{3^{2} {\sqrt{3}}^{2} + {(13)}^{2}}$

Step 5.3.1.2

Raise $3$ to the power of $2$ .

$\sqrt{9 {\sqrt{3}}^{2} + {(13)}^{2}}$

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

$\sqrt{9 {(3^{\frac{1}{2}})}^{2} + {(13)}^{2}}$

Step 5.3.2.2

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

$\sqrt{9 \cdot 3^{\frac{1}{2} \cdot 2} + {(13)}^{2}}$

Step 5.3.2.3

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

$\sqrt{9 \cdot 3^{\frac{2}{2}} + {(13)}^{2}}$

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.

$\sqrt{9 \cdot 3^{\frac{2}{2}} + {(13)}^{2}}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{9 \cdot 3^{1} + {(13)}^{2}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{9 \cdot 3 + {(13)}^{2}}$

Step 5.3.3

Simplify the expression.

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Step 5.3.3.1

Multiply $9$ by $3$ .

$\sqrt{27 + {(13)}^{2}}$

Step 5.3.3.2

Raise $13$ to the power of $2$ .

$\sqrt{27 + 169}$

Step 5.3.3.3

Add $27$ and $169$ .

$\sqrt{196}$

Step 5.3.3.4

Rewrite $196$ as $14^{2}$ .

$\sqrt{14^{2}}$

Step 5.3.3.5

Pull terms out from under the radical, assuming positive real numbers.

$14$

Step 6

Find the vertices.

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Step 6.1

The first vertex of a hyperbola 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 and simplify.

$(1 + 3 \sqrt{3}, - 2)$

Step 6.3

The second vertex of a hyperbola can be found by subtracting $a$ from $h$ .

$(h - a, k)$

Step 6.4

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

$(1 - 3 \sqrt{3}, - 2)$

Step 6.5

The vertices of a hyperbola follow the form of $(h \pm a, k)$ . Hyperbolas have two vertices.

$(1 + 3 \sqrt{3}, - 2), (1 - 3 \sqrt{3}, - 2)$

Step 7

Find the foci.

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Step 7.1

The first focus of a hyperbola 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 and simplify.

$(15, - 2)$

Step 7.3

The second focus of a hyperbola can be found by subtracting $c$ from $h$ .

$(h - c, k)$

Step 7.4

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

$(- 13, - 2)$

Step 7.5

The foci of a hyperbola follow the form of $(h \pm \sqrt{a^{2} + b^{2}}, k)$ . Hyperbolas have two foci.

$(15, - 2), (- 13, - 2)$

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.

$\frac{\sqrt{{(3 \sqrt{3})}^{2} + {(13)}^{2}}}{3 \sqrt{3}}$

Step 8.3

Simplify.

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

Apply the product rule to $3 \sqrt{3}$ .

$\frac{\sqrt{3^{2} {\sqrt{3}}^{2} + 13^{2}}}{3 \sqrt{3}}$

Step 8.3.1.2

Raise $3$ to the power of $2$ .

$\frac{\sqrt{9 {\sqrt{3}}^{2} + 13^{2}}}{3 \sqrt{3}}$

Step 8.3.1.3

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

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Step 8.3.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{9 {(3^{\frac{1}{2}})}^{2} + 13^{2}}}{3 \sqrt{3}}$

Step 8.3.1.3.2

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

$\frac{\sqrt{9 \cdot 3^{\frac{1}{2} \cdot 2} + 13^{2}}}{3 \sqrt{3}}$

Step 8.3.1.3.3

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

$\frac{\sqrt{9 \cdot 3^{\frac{2}{2}} + 13^{2}}}{3 \sqrt{3}}$

Step 8.3.1.3.4

Cancel the common factor of $2$ .

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Step 8.3.1.3.4.1

Cancel the common factor.

$\frac{\sqrt{9 \cdot 3^{\frac{2}{2}} + 13^{2}}}{3 \sqrt{3}}$

Step 8.3.1.3.4.2

Rewrite the expression.

$\frac{\sqrt{9 \cdot 3^{1} + 13^{2}}}{3 \sqrt{3}}$

Step 8.3.1.3.5

Evaluate the exponent.

$\frac{\sqrt{9 \cdot 3 + 13^{2}}}{3 \sqrt{3}}$

Step 8.3.1.4

Multiply $9$ by $3$ .

$\frac{\sqrt{27 + 13^{2}}}{3 \sqrt{3}}$

Step 8.3.1.5

Raise $13$ to the power of $2$ .

$\frac{\sqrt{27 + 169}}{3 \sqrt{3}}$

Step 8.3.1.6

Add $27$ and $169$ .

$\frac{\sqrt{196}}{3 \sqrt{3}}$

Step 8.3.1.7

Rewrite $196$ as $14^{2}$ .

$\frac{\sqrt{14^{2}}}{3 \sqrt{3}}$

Step 8.3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$\frac{14}{3 \sqrt{3}}$

Step 8.3.2

Multiply $\frac{14}{3 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{14}{3 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 8.3.3

Combine and simplify the denominator.

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Step 8.3.3.1

Multiply $\frac{14}{3 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{14 \sqrt{3}}{3 \sqrt{3} \sqrt{3}}$

Step 8.3.3.2

Move $\sqrt{3}$ .

$\frac{14 \sqrt{3}}{3 (\sqrt{3} \sqrt{3})}$

Step 8.3.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{14 \sqrt{3}}{3 ({\sqrt{3}}^{1} \sqrt{3})}$

Step 8.3.3.4

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{14 \sqrt{3}}{3 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 8.3.3.5

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

$\frac{14 \sqrt{3}}{3 {\sqrt{3}}^{1 + 1}}$

Step 8.3.3.6

Add $1$ and $1$ .

$\frac{14 \sqrt{3}}{3 {\sqrt{3}}^{2}}$

Step 8.3.3.7

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

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Step 8.3.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{14 \sqrt{3}}{3 {(3^{\frac{1}{2}})}^{2}}$

Step 8.3.3.7.2

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

$\frac{14 \sqrt{3}}{3 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 8.3.3.7.3

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

$\frac{14 \sqrt{3}}{3 \cdot 3^{\frac{2}{2}}}$

Step 8.3.3.7.4

Cancel the common factor of $2$ .

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Step 8.3.3.7.4.1

Cancel the common factor.

$\frac{14 \sqrt{3}}{3 \cdot 3^{\frac{2}{2}}}$

Step 8.3.3.7.4.2

Rewrite the expression.

$\frac{14 \sqrt{3}}{3 \cdot 3^{1}}$

Step 8.3.3.7.5

Evaluate the exponent.

$\frac{14 \sqrt{3}}{3 \cdot 3}$

Step 8.3.4

Multiply $3$ by $3$ .

$\frac{14 \sqrt{3}}{9}$

Step 9

Find the focal parameter.

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Step 9.1

Find the value of the focal parameter of the hyperbola by using the following formula.

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

Step 9.2

Substitute the values of $b$ and $\sqrt{a^{2} + b^{2}}$ in the formula.

$\frac{13^{2}}{14}$

Step 9.3

Raise $13$ to the power of $2$ .

$\frac{169}{14}$

Step 10

The asymptotes follow the form $y = \pm \frac{b (x - h)}{a} + k$ because this hyperbola opens left and right.

$y = \pm \frac{13 \sqrt{3}}{9} \cdot (x - (1)) - 2$

Step 11

Simplify to find the first asymptote.

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Step 11.1

Remove parentheses.

$y = \frac{13 \sqrt{3}}{9} \cdot (x - (1)) - 2$

Step 11.2

Simplify each term.

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Step 11.2.1

Multiply $- 1$ by $1$ .

$y = \frac{13 \sqrt{3}}{9} \cdot (x - 1) - 2$

Step 11.2.2

Apply the distributive property.

$y = \frac{13 \sqrt{3}}{9} x + \frac{13 \sqrt{3}}{9} \cdot - 1 - 2$

Step 11.2.3

Combine $\frac{13 \sqrt{3}}{9}$ and $x$ .

$y = \frac{13 \sqrt{3} x}{9} + \frac{13 \sqrt{3}}{9} \cdot - 1 - 2$

Step 11.2.4

Multiply $\frac{13 \sqrt{3}}{9} \cdot - 1$ .

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Step 11.2.4.1

Combine $\frac{13 \sqrt{3}}{9}$ and $- 1$ .

$y = \frac{13 \sqrt{3} x}{9} + \frac{13 \sqrt{3} \cdot - 1}{9} - 2$

Step 11.2.4.2

Multiply $- 1$ by $13$ .

$y = \frac{13 \sqrt{3} x}{9} + \frac{- 13 \sqrt{3}}{9} - 2$

Step 11.2.5

Move the negative in front of the fraction.

$y = \frac{13 \sqrt{3} x}{9} - \frac{13 \sqrt{3}}{9} - 2$

Step 12

Simplify to find the second asymptote.

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Step 12.1

Remove parentheses.

$y = - \frac{13 \sqrt{3}}{9} \cdot (x - (1)) - 2$

Step 12.2

Simplify each term.

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Step 12.2.1

Multiply $- 1$ by $1$ .

$y = - \frac{13 \sqrt{3}}{9} \cdot (x - 1) - 2$

Step 12.2.2

Apply the distributive property.

$y = - \frac{13 \sqrt{3}}{9} x - \frac{13 \sqrt{3}}{9} \cdot - 1 - 2$

Step 12.2.3

Combine $x$ and $\frac{13 \sqrt{3}}{9}$ .

$y = - \frac{x (13 \sqrt{3})}{9} - \frac{13 \sqrt{3}}{9} \cdot - 1 - 2$

Step 12.2.4

Multiply $- \frac{13 \sqrt{3}}{9} \cdot - 1$ .

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Step 12.2.4.1

Multiply $- 1$ by $- 1$ .

$y = - \frac{x (13 \sqrt{3})}{9} + 1 \frac{13 \sqrt{3}}{9} - 2$

Step 12.2.4.2

Multiply $\frac{13 \sqrt{3}}{9}$ by $1$ .

$y = - \frac{x (13 \sqrt{3})}{9} + \frac{13 \sqrt{3}}{9} - 2$

Step 12.2.5

Move $13$ to the left of $x$ .

$y = - \frac{13 x \sqrt{3}}{9} + \frac{13 \sqrt{3}}{9} - 2$

Step 13

This hyperbola has two asymptotes.

$y = \frac{13 \sqrt{3} x}{9} - \frac{13 \sqrt{3}}{9} - 2, y = - \frac{13 x \sqrt{3}}{9} + \frac{13 \sqrt{3}}{9} - 2$

Step 14

These values represent the important values for graphing and analyzing a hyperbola.

Center: $(1, - 2)$

Vertices: $(1 + 3 \sqrt{3}, - 2), (1 - 3 \sqrt{3}, - 2)$

Foci: $(15, - 2), (- 13, - 2)$

Eccentricity: $\frac{14 \sqrt{3}}{9}$

Focal Parameter: $\frac{169}{14}$

Asymptotes: $y = \frac{13 \sqrt{3} x}{9} - \frac{13 \sqrt{3}}{9} - 2$ , $y = - \frac{13 x \sqrt{3}}{9} + \frac{13 \sqrt{3}}{9} - 2$

Step 15