Pre-Algebra Examples

$64 y^{2} - 25 x^{2} = 1600$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Divide each term by $1600$ to make the right side equal to one.

$\frac{64 y^{2}}{1600} - \frac{25 x^{2}}{1600} = \frac{1600}{1600}$

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 $1$ .

$\frac{y^{2}}{25} - \frac{x^{2}}{64} = 1$

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, $a$ .

$a = 5$

$b = 8$

$k = 0$

$h = 0$

Step 4

The center of a hyperbola 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 hyperbola by using the following formula.

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

Step 5.2

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

$\sqrt{{(5)}^{2} + {(8)}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

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

$\sqrt{25 + {(8)}^{2}}$

Step 5.3.2

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

$\sqrt{25 + 64}$

Step 5.3.3

Add $25$ and $64$ .

$\sqrt{89}$

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 $k$ .

$(h, k + a)$

Step 6.2

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

$(0, 5)$

Step 6.3

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

$(h, k - a)$

Step 6.4

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

$(0, - 5)$

Step 6.5

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

$(0, 5), (0, - 5)$

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 $k$ .

$(h, k + c)$

Step 7.2

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

$(0, \sqrt{89})$

Step 7.3

The second focus of a hyperbola 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 and simplify.

$(0, - \sqrt{89})$

Step 7.5

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

$(0, \sqrt{89}), (0, - \sqrt{89})$

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.

$\frac{\sqrt{{(5)}^{2} + {(8)}^{2}}}{5}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

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

$\frac{\sqrt{25 + 8^{2}}}{5}$

Step 8.3.2

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

$\frac{\sqrt{25 + 64}}{5}$

Step 8.3.3

Add $25$ and $64$ .

$\frac{\sqrt{89}}{5}$

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.

$\frac{8^{2}}{\sqrt{89}}$

Step 9.3

Simplify.

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Step 9.3.1

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

$\frac{64}{\sqrt{89}}$

Step 9.3.2

Multiply $\frac{64}{\sqrt{89}}$ by $\frac{\sqrt{89}}{\sqrt{89}}$ .

$\frac{64}{\sqrt{89}} \cdot \frac{\sqrt{89}}{\sqrt{89}}$

Step 9.3.3

Combine and simplify the denominator.

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Step 9.3.3.1

Multiply $\frac{64}{\sqrt{89}}$ by $\frac{\sqrt{89}}{\sqrt{89}}$ .

$\frac{64 \sqrt{89}}{\sqrt{89} \sqrt{89}}$

Step 9.3.3.2

Raise $\sqrt{89}$ to the power of $1$ .

$\frac{64 \sqrt{89}}{{\sqrt{89}}^{1} \sqrt{89}}$

Step 9.3.3.3

Raise $\sqrt{89}$ to the power of $1$ .

$\frac{64 \sqrt{89}}{{\sqrt{89}}^{1} {\sqrt{89}}^{1}}$

Step 9.3.3.4

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

$\frac{64 \sqrt{89}}{{\sqrt{89}}^{1 + 1}}$

Step 9.3.3.5

Add $1$ and $1$ .

$\frac{64 \sqrt{89}}{{\sqrt{89}}^{2}}$

Step 9.3.3.6

Rewrite ${\sqrt{89}}^{2}$ as $89$ .

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Step 9.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{89}$ as $89^{\frac{1}{2}}$ .

$\frac{64 \sqrt{89}}{{(89^{\frac{1}{2}})}^{2}}$

Step 9.3.3.6.2

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

$\frac{64 \sqrt{89}}{89^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

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

$\frac{64 \sqrt{89}}{89^{\frac{2}{2}}}$

Step 9.3.3.6.4

Cancel the common factor of $2$ .

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Step 9.3.3.6.4.1

Cancel the common factor.

$\frac{64 \sqrt{89}}{89^{\frac{2}{2}}}$

Step 9.3.3.6.4.2

Rewrite the expression.

$\frac{64 \sqrt{89}}{89^{1}}$

Step 9.3.3.6.5

Evaluate the exponent.

$\frac{64 \sqrt{89}}{89}$

Step 10

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

$y = \pm \frac{5}{8} x + 0$

Step 11

Simplify $\frac{5}{8} x + 0$ .

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Step 11.1

Add $\frac{5}{8} x$ and $0$ .

$y = \frac{5}{8} x$

Step 11.2

Combine $\frac{5}{8}$ and $x$ .

$y = \frac{5 x}{8}$

Step 12

Simplify $- \frac{5}{8} x + 0$ .

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Step 12.1

Add $- \frac{5}{8} x$ and $0$ .

$y = - \frac{5}{8} x$

Step 12.2

Combine $x$ and $\frac{5}{8}$ .

$y = - \frac{x \cdot 5}{8}$

Step 12.3

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

$y = - \frac{5 x}{8}$

Step 13

This hyperbola has two asymptotes.

$y = \frac{5 x}{8}, y = - \frac{5 x}{8}$

Step 14

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

Center: $(0, 0)$

Vertices: $(0, 5), (0, - 5)$

Foci: $(0, \sqrt{89}), (0, - \sqrt{89})$

Eccentricity: $\frac{\sqrt{89}}{5}$

Focal Parameter: $\frac{64 \sqrt{89}}{89}$

Asymptotes: $y = \frac{5 x}{8}$ , $y = - \frac{5 x}{8}$

Step 15