Algebra Examples

$5 y^{2} - 4 x^{2} = 20$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Divide each term by $20$ to make the right side equal to one.

$\frac{5 y^{2}}{20} - \frac{4 x^{2}}{20} = \frac{20}{20}$

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 $1$ .

$\frac{y^{2}}{4} - \frac{x^{2}}{5} = 1$

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, $a$ .

$a = 2$

$b = \sqrt{5}$

$k = 0$

$h = 0$

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Simplify.

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Step 4.3.1

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

$\sqrt{4 + {(\sqrt{5})}^{2}}$

Step 4.3.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{4 + {(5^{\frac{1}{2}})}^{2}}$

Step 4.3.2.2

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

$\sqrt{4 + 5^{\frac{1}{2} \cdot 2}}$

Step 4.3.2.3

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

$\sqrt{4 + 5^{\frac{2}{2}}}$

Step 4.3.2.4

Cancel the common factor of $2$ .

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Step 4.3.2.4.1

Cancel the common factor.

$\sqrt{4 + 5^{\frac{2}{2}}}$

Step 4.3.2.4.2

Rewrite the expression.

$\sqrt{4 + 5^{1}}$

Step 4.3.2.5

Evaluate the exponent.

$\sqrt{4 + 5}$

Step 4.3.3

Simplify the expression.

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Step 4.3.3.1

Add $4$ and $5$ .

$\sqrt{9}$

Step 4.3.3.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2}}$

Step 4.3.4

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

$3$

Step 5

Find the foci.

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Step 5.1

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

$(h, k + c)$

Step 5.2

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

$(0, 3)$

Step 5.3

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

$(h, k - c)$

Step 5.4

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

$(0, - 3)$

Step 5.5

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

$(0, 3), (0, - 3)$

Step 6