Precalculus Examples

$- 25 x^{2} + 200 x + 9 y^{2} - 90 y - 400 = 0$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Add $400$ to both sides of the equation.

$- 25 x^{2} + 200 x + 9 y^{2} - 90 y = 400$

Step 1.2

Complete the square for $- 25 x^{2} + 200 x$ .

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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 $c$ .

$a = - 25$

$b = 200$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{200}{2 \cdot - 25}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $200$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $200$ .

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

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 $2 \cdot - 25$ .

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

Step 1.2.3.2.1.2.2

Cancel the common factor.

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

Step 1.2.3.2.1.2.3

Rewrite the expression.

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

Step 1.2.3.2.2

Cancel the common factor of $100$ and $- 25$ .

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Step 1.2.3.2.2.1

Factor $25$ out of $100$ .

$d = \frac{25 (4)}{- 25}$

Step 1.2.3.2.2.2

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

$d = - 1 \cdot 4$

Step 1.2.3.2.3

Multiply $- 1$ by $4$ .

$d = - 4$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{200^{2}}{4 \cdot - 25}$

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 $200$ to the power of $2$ .

$e = 0 - \frac{40000}{4 \cdot - 25}$

Step 1.2.4.2.1.2

Multiply $4$ by $- 25$ .

$e = 0 - \frac{40000}{- 100}$

Step 1.2.4.2.1.3

Divide $40000$ by $- 100$ .

$e = 0 - - 400$

Step 1.2.4.2.1.4

Multiply $- 1$ by $- 400$ .

$e = 0 + 400$

Step 1.2.4.2.2

Add $0$ and $400$ .

$e = 400$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 25 {(x - 4)}^{2} + 400$ .

$- 25 {(x - 4)}^{2} + 400$

Step 1.3

Substitute $- 25 {(x - 4)}^{2} + 400$ for $- 25 x^{2} + 200 x$ in the equation $- 25 x^{2} + 200 x + 9 y^{2} - 90 y = 400$ .

$- 25 {(x - 4)}^{2} + 400 + 9 y^{2} - 90 y = 400$

Step 1.4

Move $400$ to the right side of the equation by adding $400$ to both sides.

$- 25 {(x - 4)}^{2} + 9 y^{2} - 90 y = 400 - 400$

Step 1.5

Complete the square for $9 y^{2} - 90 y$ .

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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 $c$ .

$a = 9$

$b = - 90$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 90}{2 \cdot 9}$

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Cancel the common factor of $- 90$ and $2$ .

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Step 1.5.3.2.1.1

Factor $2$ out of $- 90$ .

$d = \frac{2 \cdot - 45}{2 \cdot 9}$

Step 1.5.3.2.1.2

Cancel the common factors.

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Step 1.5.3.2.1.2.1

Factor $2$ out of $2 \cdot 9$ .

$d = \frac{2 \cdot - 45}{2 (9)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 45}{2 \cdot 9}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 45}{9}$

Step 1.5.3.2.2

Cancel the common factor of $- 45$ and $9$ .

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Step 1.5.3.2.2.1

Factor $9$ out of $- 45$ .

$d = \frac{9 \cdot - 5}{9}$

Step 1.5.3.2.2.2

Cancel the common factors.

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Step 1.5.3.2.2.2.1

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 5}{9 (1)}$

Step 1.5.3.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot - 5}{9 \cdot 1}$

Step 1.5.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 5}{1}$

Step 1.5.3.2.2.2.4

Divide $- 5$ by $1$ .

$d = - 5$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 90)}^{2}}{4 \cdot 9}$

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

Raise $- 90$ to the power of $2$ .

$e = 0 - \frac{8100}{4 \cdot 9}$

Step 1.5.4.2.1.2

Multiply $4$ by $9$ .

$e = 0 - \frac{8100}{36}$

Step 1.5.4.2.1.3

Divide $8100$ by $36$ .

$e = 0 - 1 \cdot 225$

Step 1.5.4.2.1.4

Multiply $- 1$ by $225$ .

$e = 0 - 225$

Step 1.5.4.2.2

Subtract $225$ from $0$ .

$e = - 225$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(y - 5)}^{2} - 225$ .

$9 {(y - 5)}^{2} - 225$

Step 1.6

Substitute $9 {(y - 5)}^{2} - 225$ for $9 y^{2} - 90 y$ in the equation $- 25 x^{2} + 200 x + 9 y^{2} - 90 y = 400$ .

$- 25 {(x - 4)}^{2} + 9 {(y - 5)}^{2} - 225 = 400 - 400$

Step 1.7

Move $- 225$ to the right side of the equation by adding $225$ to both sides.

$- 25 {(x - 4)}^{2} + 9 {(y - 5)}^{2} = 400 - 400 + 225$

Step 1.8

Simplify $400 - 400 + 225$ .

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Step 1.8.1

Subtract $400$ from $400$ .

$- 25 {(x - 4)}^{2} + 9 {(y - 5)}^{2} = 0 + 225$

Step 1.8.2

Add $0$ and $225$ .

$- 25 {(x - 4)}^{2} + 9 {(y - 5)}^{2} = 225$

Step 1.9

Divide each term by $225$ to make the right side equal to one.

$- \frac{25 {(x - 4)}^{2}}{225} + \frac{9 {(y - 5)}^{2}}{225} = \frac{225}{225}$

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 $1$ .

$\frac{{(y - 5)}^{2}}{25} - \frac{{(x - 4)}^{2}}{9} = 1$

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, $a$ .

$a = 5$

$b = 3$

$k = 5$

$h = 4$

Step 4

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

$y = \pm \frac{5}{3} \cdot (x - (4)) + 5$

Step 5

Simplify to find the first asymptote.

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Step 5.1

Remove parentheses.

$y = \frac{5}{3} \cdot (x - (4)) + 5$

Step 5.2

Simplify $\frac{5}{3} \cdot (x - (4)) + 5$ .

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $- 1$ by $4$ .

$y = \frac{5}{3} \cdot (x - 4) + 5$

Step 5.2.1.2

Apply the distributive property.

$y = \frac{5}{3} x + \frac{5}{3} \cdot - 4 + 5$

Step 5.2.1.3

Combine $\frac{5}{3}$ and $x$ .

$y = \frac{5 x}{3} + \frac{5}{3} \cdot - 4 + 5$

Step 5.2.1.4

Multiply $\frac{5}{3} \cdot - 4$ .

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Step 5.2.1.4.1

Combine $\frac{5}{3}$ and $- 4$ .

$y = \frac{5 x}{3} + \frac{5 \cdot - 4}{3} + 5$

Step 5.2.1.4.2

Multiply $5$ by $- 4$ .

$y = \frac{5 x}{3} + \frac{- 20}{3} + 5$

Step 5.2.1.5

Move the negative in front of the fraction.

$y = \frac{5 x}{3} - \frac{20}{3} + 5$

Step 5.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{5 x}{3} - \frac{20}{3} + 5 \cdot \frac{3}{3}$

Step 5.2.3

Combine $5$ and $\frac{3}{3}$ .

$y = \frac{5 x}{3} - \frac{20}{3} + \frac{5 \cdot 3}{3}$

Step 5.2.4

Combine the numerators over the common denominator.

$y = \frac{5 x}{3} + \frac{- 20 + 5 \cdot 3}{3}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $5$ by $3$ .

$y = \frac{5 x}{3} + \frac{- 20 + 15}{3}$

Step 5.2.5.2

Add $- 20$ and $15$ .

$y = \frac{5 x}{3} + \frac{- 5}{3}$

Step 5.2.6

Move the negative in front of the fraction.

$y = \frac{5 x}{3} - \frac{5}{3}$

Step 6

Simplify to find the second asymptote.

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Step 6.1

Remove parentheses.

$y = - \frac{5}{3} \cdot (x - (4)) + 5$

Step 6.2

Simplify $- \frac{5}{3} \cdot (x - (4)) + 5$ .

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $- 1$ by $4$ .

$y = - \frac{5}{3} \cdot (x - 4) + 5$

Step 6.2.1.2

Apply the distributive property.

$y = - \frac{5}{3} x - \frac{5}{3} \cdot - 4 + 5$

Step 6.2.1.3

Combine $x$ and $\frac{5}{3}$ .

$y = - \frac{x \cdot 5}{3} - \frac{5}{3} \cdot - 4 + 5$

Step 6.2.1.4

Multiply $- \frac{5}{3} \cdot - 4$ .

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Step 6.2.1.4.1

Multiply $- 4$ by $- 1$ .

$y = - \frac{x \cdot 5}{3} + 4 (\frac{5}{3}) + 5$

Step 6.2.1.4.2

Combine $4$ and $\frac{5}{3}$ .

$y = - \frac{x \cdot 5}{3} + \frac{4 \cdot 5}{3} + 5$

Step 6.2.1.4.3

Multiply $4$ by $5$ .

$y = - \frac{x \cdot 5}{3} + \frac{20}{3} + 5$

Step 6.2.1.5

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

$y = - \frac{5 x}{3} + \frac{20}{3} + 5$

Step 6.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{5 x}{3} + \frac{20}{3} + 5 \cdot \frac{3}{3}$

Step 6.2.3

Combine $5$ and $\frac{3}{3}$ .

$y = - \frac{5 x}{3} + \frac{20}{3} + \frac{5 \cdot 3}{3}$

Step 6.2.4

Combine the numerators over the common denominator.

$y = - \frac{5 x}{3} + \frac{20 + 5 \cdot 3}{3}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $5$ by $3$ .

$y = - \frac{5 x}{3} + \frac{20 + 15}{3}$

Step 6.2.5.2

Add $20$ and $15$ .

$y = - \frac{5 x}{3} + \frac{35}{3}$

Step 7

This hyperbola has two asymptotes.

$y = \frac{5 x}{3} - \frac{5}{3}, y = - \frac{5 x}{3} + \frac{35}{3}$

Step 8

The asymptotes are $y = \frac{5 x}{3} - \frac{5}{3}$ and $y = - \frac{5 x}{3} + \frac{35}{3}$ .

Asymptotes: $y = \frac{5 x}{3} - \frac{5}{3}, y = - \frac{5 x}{3} + \frac{35}{3}$

Step 9