Trigonometry Examples

$4 y^{2} - x^{2} - 24 y - 4 x + 16 = 0$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Subtract $16$ from both sides of the equation.

$4 y^{2} - x^{2} - 24 y - 4 x = - 16$

Step 1.2

Complete the square for $4 y^{2} - 24 y$ .

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Step 1.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 4$

$b = - 24$

$c = 0$

Step 1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 24}{2 \cdot 4}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 24$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 24$ .

$d = \frac{2 \cdot - 12}{2 \cdot 4}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot - 12}{2 (4)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 12}{2 \cdot 4}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 12}{4}$

Step 1.2.3.2.2

Cancel the common factor of $- 12$ and $4$ .

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Step 1.2.3.2.2.1

Factor $4$ out of $- 12$ .

$d = \frac{4 \cdot - 3}{4}$

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $4$ out of $4$ .

$d = \frac{4 \cdot - 3}{4 (1)}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{4 \cdot - 3}{4 \cdot 1}$

Step 1.2.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 3}{1}$

Step 1.2.3.2.2.2.4

Divide $- 3$ by $1$ .

$d = - 3$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 24)}^{2}}{4 \cdot 4}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Raise $- 24$ to the power of $2$ .

$e = 0 - \frac{576}{4 \cdot 4}$

Step 1.2.4.2.1.2

Multiply $4$ by $4$ .

$e = 0 - \frac{576}{16}$

Step 1.2.4.2.1.3

Divide $576$ by $16$ .

$e = 0 - 1 \cdot 36$

Step 1.2.4.2.1.4

Multiply $- 1$ by $36$ .

$e = 0 - 36$

Step 1.2.4.2.2

Subtract $36$ from $0$ .

$e = - 36$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(y - 3)}^{2} - 36$ .

$4 {(y - 3)}^{2} - 36$

Step 1.3

Substitute $4 {(y - 3)}^{2} - 36$ for $4 y^{2} - 24 y$ in the equation $4 y^{2} - x^{2} - 24 y - 4 x = - 16$ .

$4 {(y - 3)}^{2} - 36 - x^{2} - 4 x = - 16$

Step 1.4

Move $- 36$ to the right side of the equation by adding $36$ to both sides.

$4 {(y - 3)}^{2} - x^{2} - 4 x = - 16 + 36$

Step 1.5

Complete the square for $- x^{2} - 4 x$ .

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Step 1.5.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 1$

$b = - 4$

$c = 0$

Step 1.5.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.5.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.5.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 4}{2 \cdot - 1}$

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Cancel the common factor of $- 4$ and $2$ .

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Step 1.5.3.2.1.1

Factor $2$ out of $- 4$ .

$d = \frac{2 \cdot - 2}{2 \cdot - 1}$

Step 1.5.3.2.1.2

Move the negative one from the denominator of $\frac{- 2}{- 1}$ .

$d = - 1 \cdot - 2$

Step 1.5.3.2.2

Rewrite $- 1 \cdot - 2$ as $- - 2$ .

$d = - - 2$

Step 1.5.3.2.3

Multiply $- 1$ by $- 2$ .

$d = 2$

Step 1.5.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.5.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 4)}^{2}}{4 \cdot - 1}$

Step 1.5.4.2

Simplify the right side.

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Step 1.5.4.2.1

Simplify each term.

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Step 1.5.4.2.1.1

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

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Step 1.5.4.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

$e = 0 - \frac{{(- 1 (4))}^{2}}{4 \cdot - 1}$

Step 1.5.4.2.1.1.2

Apply the product rule to $- 1 (4)$ .

$e = 0 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot - 1}$

Step 1.5.4.2.1.1.3

Raise $- 1$ to the power of $2$ .

$e = 0 - \frac{1 \cdot 4^{2}}{4 \cdot - 1}$

Step 1.5.4.2.1.1.4

Multiply $4^{2}$ by $1$ .

$e = 0 - \frac{4^{2}}{4 \cdot - 1}$

Step 1.5.4.2.1.1.5

Factor $4$ out of $4^{2}$ .

$e = 0 - \frac{4 \cdot 4}{4 \cdot - 1}$

Step 1.5.4.2.1.1.6

Move the negative one from the denominator of $\frac{4}{- 1}$ .

$e = 0 - (- 1 \cdot 4)$

Step 1.5.4.2.1.2

Multiply $- (- 1 \cdot 4)$ .

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Step 1.5.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 0 - - 4$

Step 1.5.4.2.1.2.2

Multiply $- 1$ by $- 4$ .

$e = 0 + 4$

Step 1.5.4.2.2

Add $0$ and $4$ .

$e = 4$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + 2)}^{2} + 4$ .

$- {(x + 2)}^{2} + 4$

Step 1.6

Substitute $- {(x + 2)}^{2} + 4$ for $- x^{2} - 4 x$ in the equation $4 y^{2} - x^{2} - 24 y - 4 x = - 16$ .

$4 {(y - 3)}^{2} - {(x + 2)}^{2} + 4 = - 16 + 36$

Step 1.7

Move $4$ to the right side of the equation by adding $4$ to both sides.

$4 {(y - 3)}^{2} - {(x + 2)}^{2} = - 16 + 36 - 4$

Step 1.8

Simplify $- 16 + 36 - 4$ .

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Step 1.8.1

Add $- 16$ and $36$ .

$4 {(y - 3)}^{2} - {(x + 2)}^{2} = 20 - 4$

Step 1.8.2

Subtract $4$ from $20$ .

$4 {(y - 3)}^{2} - {(x + 2)}^{2} = 16$

Step 1.9

Divide each term by $16$ to make the right side equal to one.

$\frac{4 {(y - 3)}^{2}}{16} - \frac{{(x + 2)}^{2}}{16} = \frac{16}{16}$

Step 1.10

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 $1$ .

$\frac{{(y - 3)}^{2}}{4} - \frac{{(x + 2)}^{2}}{16} = 1$

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, $a$ .

$a = 2$

$b = 4$

$k = 3$

$h = - 2$

Step 4

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

$(- 2, 3)$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

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

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

Step 5.3.2

Raise $4$ to the power of $2$ .

$\sqrt{4 + 16}$

Step 5.3.3

Add $4$ and $16$ .

$\sqrt{20}$

Step 5.3.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 5.3.4.1

Factor $4$ out of $20$ .

$\sqrt{4 (5)}$

Step 5.3.4.2

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 5}$

Step 5.3.5

Pull terms out from under the radical.

$2 \sqrt{5}$

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.

$(- 2, 5)$

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.

$(- 2, 1)$

Step 6.5

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

$(- 2, 5), (- 2, 1)$

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.

$(- 2, 3 + 2 \sqrt{5})$

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.

$(- 2, 3 - 2 \sqrt{5})$

Step 7.5

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

$(- 2, 3 + 2 \sqrt{5}), (- 2, 3 - 2 \sqrt{5})$

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.

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

Step 8.3

Simplify.

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

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

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

Step 8.3.1.2

Raise $4$ to the power of $2$ .

$\frac{\sqrt{4 + 16}}{2}$

Step 8.3.1.3

Add $4$ and $16$ .

$\frac{\sqrt{20}}{2}$

Step 8.3.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 8.3.1.4.1

Factor $4$ out of $20$ .

$\frac{\sqrt{4 (5)}}{2}$

Step 8.3.1.4.2

Rewrite $4$ as $2^{2}$ .

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

Step 8.3.1.5

Pull terms out from under the radical.

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

Step 8.3.2

Cancel the common factor of $2$ .

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Step 8.3.2.1

Cancel the common factor.

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

Step 8.3.2.2

Divide $\sqrt{5}$ by $1$ .

$\sqrt{5}$

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.

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

Step 9.3

Simplify.

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Step 9.3.1

Raise $4$ to the power of $2$ .

$\frac{16}{2 \sqrt{5}}$

Step 9.3.2

Cancel the common factor of $16$ and $2$ .

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Step 9.3.2.1

Factor $2$ out of $16$ .

$\frac{2 \cdot 8}{2 \sqrt{5}}$

Step 9.3.2.2

Cancel the common factors.

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Step 9.3.2.2.1

Factor $2$ out of $2 \sqrt{5}$ .

$\frac{2 \cdot 8}{2 (\sqrt{5})}$

Step 9.3.2.2.2

Cancel the common factor.

$\frac{2 \cdot 8}{2 \sqrt{5}}$

Step 9.3.2.2.3

Rewrite the expression.

$\frac{8}{\sqrt{5}}$

Step 9.3.3

Multiply $\frac{8}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{8}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 9.3.4

Combine and simplify the denominator.

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Step 9.3.4.1

Multiply $\frac{8}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{8 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 9.3.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{8 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 9.3.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{8 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 9.3.4.4

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

$\frac{8 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 9.3.4.5

Add $1$ and $1$ .

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

Step 9.3.4.6

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

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Step 9.3.4.6.1

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

$\frac{8 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 9.3.4.6.2

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

$\frac{8 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 9.3.4.6.3

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

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

Step 9.3.4.6.4

Cancel the common factor of $2$ .

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Step 9.3.4.6.4.1

Cancel the common factor.

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

Step 9.3.4.6.4.2

Rewrite the expression.

$\frac{8 \sqrt{5}}{5^{1}}$

Step 9.3.4.6.5

Evaluate the exponent.

$\frac{8 \sqrt{5}}{5}$

Step 10

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

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

Step 11

Simplify to find the first asymptote.

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Step 11.1

Remove parentheses.

$y = \frac{1}{2} \cdot (x - (- 2)) + 3$

Step 11.2

Simplify $\frac{1}{2} \cdot (x - (- 2)) + 3$ .

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Multiply $- 1$ by $- 2$ .

$y = \frac{1}{2} \cdot (x + 2) + 3$

Step 11.2.1.2

Apply the distributive property.

$y = \frac{1}{2} x + \frac{1}{2} \cdot 2 + 3$

Step 11.2.1.3

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

$y = \frac{x}{2} + \frac{1}{2} \cdot 2 + 3$

Step 11.2.1.4

Cancel the common factor of $2$ .

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Step 11.2.1.4.1

Cancel the common factor.

$y = \frac{x}{2} + \frac{1}{2} \cdot 2 + 3$

Step 11.2.1.4.2

Rewrite the expression.

$y = \frac{x}{2} + 1 + 3$

Step 11.2.2

Add $1$ and $3$ .

$y = \frac{x}{2} + 4$

Step 12

Simplify to find the second asymptote.

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Step 12.1

Remove parentheses.

$y = - \frac{1}{2} \cdot (x - (- 2)) + 3$

Step 12.2

Simplify $- \frac{1}{2} \cdot (x - (- 2)) + 3$ .

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Multiply $- 1$ by $- 2$ .

$y = - \frac{1}{2} \cdot (x + 2) + 3$

Step 12.2.1.2

Apply the distributive property.

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

Step 12.2.1.3

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

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

Step 12.2.1.4

Cancel the common factor of $2$ .

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Step 12.2.1.4.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 12.2.1.4.2

Cancel the common factor.

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

Step 12.2.1.4.3

Rewrite the expression.

$y = - \frac{x}{2} - 1 + 3$

Step 12.2.2

Add $- 1$ and $3$ .

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

Step 13

This hyperbola has two asymptotes.

$y = \frac{x}{2} + 4, y = - \frac{x}{2} + 2$

Step 14

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

Center: $(- 2, 3)$

Vertices: $(- 2, 5), (- 2, 1)$

Foci: $(- 2, 3 + 2 \sqrt{5}), (- 2, 3 - 2 \sqrt{5})$

Eccentricity: $\sqrt{5}$

Focal Parameter: $\frac{8 \sqrt{5}}{5}$

Asymptotes: $y = \frac{x}{2} + 4$ , $y = - \frac{x}{2} + 2$

Step 15