Trigonometry Examples

$4 y^{2} - 8 y - 25 x^{2} + 100 x = 196$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Complete the square for $4 y^{2} - 8 y$ .

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Step 1.1.1

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

$a = 4$

$b = - 8$

$c = 0$

Step 1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.3

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

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Step 1.1.3.1

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

$d = \frac{- 8}{2 \cdot 4}$

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Cancel the common factor of $- 8$ and $2$ .

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Step 1.1.3.2.1.1

Factor $2$ out of $- 8$ .

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

Step 1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.3.2.1.2.1

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

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

Step 1.1.3.2.1.2.2

Cancel the common factor.

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

Step 1.1.3.2.1.2.3

Rewrite the expression.

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

Step 1.1.3.2.2

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

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Step 1.1.3.2.2.1

Factor $4$ out of $- 4$ .

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

Step 1.1.3.2.2.2

Cancel the common factors.

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Step 1.1.3.2.2.2.1

Factor $4$ out of $4$ .

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

Step 1.1.3.2.2.2.2

Cancel the common factor.

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

Step 1.1.3.2.2.2.3

Rewrite the expression.

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

Step 1.1.3.2.2.2.4

Divide $- 1$ by $1$ .

$d = - 1$

Step 1.1.4

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

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Step 1.1.4.1

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

$e = 0 - \frac{{(- 8)}^{2}}{4 \cdot 4}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

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

$e = 0 - \frac{64}{4 \cdot 4}$

Step 1.1.4.2.1.2

Multiply $4$ by $4$ .

$e = 0 - \frac{64}{16}$

Step 1.1.4.2.1.3

Divide $64$ by $16$ .

$e = 0 - 1 \cdot 4$

Step 1.1.4.2.1.4

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Step 1.1.4.2.2

Subtract $4$ from $0$ .

$e = - 4$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(y - 1)}^{2} - 4$ .

$4 {(y - 1)}^{2} - 4$

Step 1.2

Substitute $4 {(y - 1)}^{2} - 4$ for $4 y^{2} - 8 y$ in the equation $4 y^{2} - 8 y - 25 x^{2} + 100 x = 196$ .

$4 {(y - 1)}^{2} - 4 - 25 x^{2} + 100 x = 196$

Step 1.3

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

$4 {(y - 1)}^{2} - 25 x^{2} + 100 x = 196 + 4$

Step 1.4

Complete the square for $- 25 x^{2} + 100 x$ .

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Step 1.4.1

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

$a = - 25$

$b = 100$

$c = 0$

Step 1.4.2

Consider the vertex form of a parabola.

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

Step 1.4.3

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

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Step 1.4.3.1

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

$d = \frac{100}{2 \cdot - 25}$

Step 1.4.3.2

Simplify the right side.

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Step 1.4.3.2.1

Cancel the common factor of $100$ and $2$ .

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Step 1.4.3.2.1.1

Factor $2$ out of $100$ .

$d = \frac{2 \cdot 50}{2 \cdot - 25}$

Step 1.4.3.2.1.2

Cancel the common factors.

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Step 1.4.3.2.1.2.1

Factor $2$ out of $2 \cdot - 25$ .

$d = \frac{2 \cdot 50}{2 (- 25)}$

Step 1.4.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 50}{2 \cdot - 25}$

Step 1.4.3.2.1.2.3

Rewrite the expression.

$d = \frac{50}{- 25}$

Step 1.4.3.2.2

Cancel the common factor of $50$ and $- 25$ .

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Step 1.4.3.2.2.1

Factor $25$ out of $50$ .

$d = \frac{25 (2)}{- 25}$

Step 1.4.3.2.2.2

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

$d = - 1 \cdot 2$

Step 1.4.3.2.3

Multiply $- 1$ by $2$ .

$d = - 2$

Step 1.4.4

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

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Step 1.4.4.1

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

$e = 0 - \frac{100^{2}}{4 \cdot - 25}$

Step 1.4.4.2

Simplify the right side.

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Step 1.4.4.2.1

Simplify each term.

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Step 1.4.4.2.1.1

Raise $100$ to the power of $2$ .

$e = 0 - \frac{10000}{4 \cdot - 25}$

Step 1.4.4.2.1.2

Multiply $4$ by $- 25$ .

$e = 0 - \frac{10000}{- 100}$

Step 1.4.4.2.1.3

Divide $10000$ by $- 100$ .

$e = 0 - - 100$

Step 1.4.4.2.1.4

Multiply $- 1$ by $- 100$ .

$e = 0 + 100$

Step 1.4.4.2.2

Add $0$ and $100$ .

$e = 100$

Step 1.4.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 25 {(x - 2)}^{2} + 100$ .

$- 25 {(x - 2)}^{2} + 100$

Step 1.5

Substitute $- 25 {(x - 2)}^{2} + 100$ for $- 25 x^{2} + 100 x$ in the equation $4 y^{2} - 8 y - 25 x^{2} + 100 x = 196$ .

$4 {(y - 1)}^{2} - 25 {(x - 2)}^{2} + 100 = 196 + 4$

Step 1.6

Move $100$ to the right side of the equation by adding $100$ to both sides.

$4 {(y - 1)}^{2} - 25 {(x - 2)}^{2} = 196 + 4 - 100$

Step 1.7

Simplify $196 + 4 - 100$ .

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Step 1.7.1

Add $196$ and $4$ .

$4 {(y - 1)}^{2} - 25 {(x - 2)}^{2} = 200 - 100$

Step 1.7.2

Subtract $100$ from $200$ .

$4 {(y - 1)}^{2} - 25 {(x - 2)}^{2} = 100$

Step 1.8

Divide each term by $100$ to make the right side equal to one.

$\frac{4 {(y - 1)}^{2}}{100} - \frac{25 {(x - 2)}^{2}}{100} = \frac{100}{100}$

Step 1.9

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 $1$ .

$\frac{{(y - 1)}^{2}}{25} - \frac{{(x - 2)}^{2}}{4} = 1$

Step 2

This is the form of a hyperbola. Use this form to determine the values used to find the 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, $a$ .

$a = 5$

$b = 2$

$k = 1$

$h = 2$

Step 4

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

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

Step 5

Simplify to find the first asymptote.

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Step 5.1

Remove parentheses.

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

Step 5.2

Simplify $\frac{5}{2} \cdot (x - (2)) + 1$ .

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $- 1$ by $2$ .

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

Step 5.2.1.2

Apply the distributive property.

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

Step 5.2.1.3

Combine $\frac{5}{2}$ and $x$ .

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

Step 5.2.1.4

Cancel the common factor of $2$ .

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Step 5.2.1.4.1

Factor $2$ out of $- 2$ .

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

Step 5.2.1.4.2

Cancel the common factor.

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

Step 5.2.1.4.3

Rewrite the expression.

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

Step 5.2.1.5

Multiply $5$ by $- 1$ .

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

Step 5.2.2

Add $- 5$ and $1$ .

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

Step 6

Simplify to find the second asymptote.

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Step 6.1

Remove parentheses.

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

Step 6.2

Simplify $- \frac{5}{2} \cdot (x - (2)) + 1$ .

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $- 1$ by $2$ .

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

Step 6.2.1.2

Apply the distributive property.

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

Step 6.2.1.3

Combine $x$ and $\frac{5}{2}$ .

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

Step 6.2.1.4

Cancel the common factor of $2$ .

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Step 6.2.1.4.1

Move the leading negative in $- \frac{5}{2}$ into the numerator.

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

Step 6.2.1.4.2

Factor $2$ out of $- 2$ .

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

Step 6.2.1.4.3

Cancel the common factor.

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

Step 6.2.1.4.4

Rewrite the expression.

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

Step 6.2.1.5

Multiply $- 5$ by $- 1$ .

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

Step 6.2.1.6

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

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

Step 6.2.2

Add $5$ and $1$ .

$y = - \frac{5 x}{2} + 6$

Step 7

This hyperbola has two asymptotes.

$y = \frac{5 x}{2} - 4, y = - \frac{5 x}{2} + 6$

Step 8

The asymptotes are $y = \frac{5 x}{2} - 4$ and $y = - \frac{5 x}{2} + 6$ .

Asymptotes: $y = \frac{5 x}{2} - 4, y = - \frac{5 x}{2} + 6$

Step 9