Algebra Examples

$\frac{y^{2}}{25} - \frac{{(x - 6)}^{2}}{144} = 1$

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 $1$ .

$\frac{y^{2}}{25} - \frac{{(x - 6)}^{2}}{144} = 1$

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.

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

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, $a$ .

$a = 5$

$b = 12$

$k = 0$

$h = 6$

Step 4

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

$(6, 0)$

Step 5

Find $c$ , the distance from the center to a focus.

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Find the distance from the center to a focus of the hyperbola by using the following formula.

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

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

$\sqrt{{(5)}^{2} + {(12)}^{2}}$

Simplify.

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Raise $5$ to the power of $2$ .

$\sqrt{25 + {(12)}^{2}}$

Raise $12$ to the power of $2$ .

$\sqrt{25 + 144}$

Add $25$ and $144$ .

$\sqrt{169}$

Rewrite $169$ as $13^{2}$ .

$\sqrt{13^{2}}$

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

$13$

Step 6

Find the vertices.

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The first vertex of a hyperbola can be found by adding $a$ to $k$ .

$(h, k + a)$

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

$(6, 5)$

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

$(h, k - a)$

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

$(6, - 5)$

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

$(6, 5), (6, - 5)$

Step 7

Find the foci.

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The first focus of a hyperbola can be found by adding $c$ to $k$ .

$(h, k + c)$

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

$(6, 13)$

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

$(h, k - c)$

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

$(6, - 13)$

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

$(6, 13), (6, - 13)$

Step 8

Find the eccentricity.

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Find the eccentricity by using the following formula.

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

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

$\frac{\sqrt{{(5)}^{2} + {(12)}^{2}}}{5}$

Simplify the numerator.

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Raise $5$ to the power of $2$ .

$\frac{\sqrt{25 + 12^{2}}}{5}$

Raise $12$ to the power of $2$ .

$\frac{\sqrt{25 + 144}}{5}$

Add $25$ and $144$ .

$\frac{\sqrt{169}}{5}$

Rewrite $169$ as $13^{2}$ .

$\frac{\sqrt{13^{2}}}{5}$

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

$\frac{13}{5}$

Step 9

Find the focal parameter.

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

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

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

$\frac{12^{2}}{13}$

Raise $12$ to the power of $2$ .

$\frac{144}{13}$

Step 10

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

$y = \pm \frac{5}{12} \cdot (x - (6)) + 0$

Step 11

Simplify to find the first asymptote.

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Remove parentheses.

$y = \frac{5}{12} \cdot (x - (6)) + 0$

Simplify $\frac{5}{12} \cdot (x - (6)) + 0$ .

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Simplify the expression.

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Add $\frac{5}{12} \cdot (x - (6))$ and $0$ .

$y = \frac{5}{12} \cdot (x - (6))$

Multiply $- 1$ by $6$ .

$y = \frac{5}{12} \cdot (x - 6)$

Apply the distributive property.

$y = \frac{5}{12} x + \frac{5}{12} \cdot - 6$

Combine $\frac{5}{12}$ and $x$ .

$y = \frac{5 x}{12} + \frac{5}{12} \cdot - 6$

Cancel the common factor of $6$ .

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Factor $6$ out of $12$ .

$y = \frac{5 x}{12} + \frac{5}{6 (2)} \cdot - 6$

Factor $6$ out of $- 6$ .

$y = \frac{5 x}{12} + \frac{5}{6 \cdot 2} \cdot (6 \cdot - 1)$

Cancel the common factor.

$y = \frac{5 x}{12} + \frac{5}{6 \cdot 2} \cdot (6 \cdot - 1)$

Rewrite the expression.

$y = \frac{5 x}{12} + \frac{5}{2} \cdot - 1$

Combine $\frac{5}{2}$ and $- 1$ .

$y = \frac{5 x}{12} + \frac{5 \cdot - 1}{2}$

Simplify the expression.

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Multiply $5$ by $- 1$ .

$y = \frac{5 x}{12} + \frac{- 5}{2}$

Move the negative in front of the fraction.

$y = \frac{5 x}{12} - \frac{5}{2}$

Step 12

Simplify to find the second asymptote.

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Remove parentheses.

$y = - \frac{5}{12} \cdot (x - (6)) + 0$

Simplify $- \frac{5}{12} \cdot (x - (6)) + 0$ .

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Simplify terms.

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Add $- \frac{5}{12} \cdot (x - (6))$ and $0$ .

$y = - \frac{5}{12} \cdot (x - (6))$

Multiply $- 1$ by $6$ .

$y = - \frac{5}{12} \cdot (x - 6)$

Apply the distributive property.

$y = - \frac{5}{12} x - \frac{5}{12} \cdot - 6$

Combine $x$ and $\frac{5}{12}$ .

$y = - \frac{x \cdot 5}{12} - \frac{5}{12} \cdot - 6$

Cancel the common factor of $6$ .

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Move the leading negative in $- \frac{5}{12}$ into the numerator.

$y = - \frac{x \cdot 5}{12} + \frac{- 5}{12} \cdot - 6$

Factor $6$ out of $12$ .

$y = - \frac{x \cdot 5}{12} + \frac{- 5}{6 (2)} \cdot - 6$

Factor $6$ out of $- 6$ .

$y = - \frac{x \cdot 5}{12} + \frac{- 5}{6 \cdot 2} \cdot (6 \cdot - 1)$

Cancel the common factor.

$y = - \frac{x \cdot 5}{12} + \frac{- 5}{6 \cdot 2} \cdot (6 \cdot - 1)$

Rewrite the expression.

$y = - \frac{x \cdot 5}{12} + \frac{- 5}{2} \cdot - 1$

Combine $\frac{- 5}{2}$ and $- 1$ .

$y = - \frac{x \cdot 5}{12} + \frac{- 5 \cdot - 1}{2}$

Multiply $- 5$ by $- 1$ .

$y = - \frac{x \cdot 5}{12} + \frac{5}{2}$

Move $5$ to the left of $x$ .

$y = - \frac{5 x}{12} + \frac{5}{2}$

Step 13

This hyperbola has two asymptotes.

$y = \frac{5 x}{12} - \frac{5}{2}, y = - \frac{5 x}{12} + \frac{5}{2}$

Step 14

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

Center: $(6, 0)$

Vertices: $(6, 5), (6, - 5)$

Foci: $(6, 13), (6, - 13)$

Eccentricity: $\frac{13}{5}$

Focal Parameter: $\frac{144}{13}$

Asymptotes: $y = \frac{5 x}{12} - \frac{5}{2}$ , $y = - \frac{5 x}{12} + \frac{5}{2}$

Step 15