Precalculus Examples

$- 16 y^{2} - 54 x + 9 x^{2} = 63$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Complete the square for $- 54 x + 9 x^{2}$ .

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Step 1.1.1

Reorder $- 54 x$ and $9 x^{2}$ .

$9 x^{2} - 54 x$

Step 1.1.2

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

$a = 9$

$b = - 54$

$c = 0$

Step 1.1.3

Consider the vertex form of a parabola.

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

Step 1.1.4

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

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Step 1.1.4.1

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

$d = \frac{- 54}{2 \cdot 9}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Cancel the common factor of $- 54$ and $2$ .

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Step 1.1.4.2.1.1

Factor $2$ out of $- 54$ .

$d = \frac{2 \cdot - 27}{2 \cdot 9}$

Step 1.1.4.2.1.2

Cancel the common factors.

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Step 1.1.4.2.1.2.1

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

$d = \frac{2 \cdot - 27}{2 (9)}$

Step 1.1.4.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 27}{2 \cdot 9}$

Step 1.1.4.2.1.2.3

Rewrite the expression.

$d = \frac{- 27}{9}$

Step 1.1.4.2.2

Cancel the common factor of $- 27$ and $9$ .

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Step 1.1.4.2.2.1

Factor $9$ out of $- 27$ .

$d = \frac{9 \cdot - 3}{9}$

Step 1.1.4.2.2.2

Cancel the common factors.

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Step 1.1.4.2.2.2.1

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 3}{9 (1)}$

Step 1.1.4.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot - 3}{9 \cdot 1}$

Step 1.1.4.2.2.2.3

Rewrite the expression.

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

Step 1.1.4.2.2.2.4

Divide $- 3$ by $1$ .

$d = - 3$

Step 1.1.5

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

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Step 1.1.5.1

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

$e = 0 - \frac{{(- 54)}^{2}}{4 \cdot 9}$

Step 1.1.5.2

Simplify the right side.

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Step 1.1.5.2.1

Simplify each term.

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Step 1.1.5.2.1.1

Raise $- 54$ to the power of $2$ .

$e = 0 - \frac{2916}{4 \cdot 9}$

Step 1.1.5.2.1.2

Multiply $4$ by $9$ .

$e = 0 - \frac{2916}{36}$

Step 1.1.5.2.1.3

Divide $2916$ by $36$ .

$e = 0 - 1 \cdot 81$

Step 1.1.5.2.1.4

Multiply $- 1$ by $81$ .

$e = 0 - 81$

Step 1.1.5.2.2

Subtract $81$ from $0$ .

$e = - 81$

Step 1.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - 3)}^{2} - 81$ .

$9 {(x - 3)}^{2} - 81$

Step 1.2

Substitute $9 {(x - 3)}^{2} - 81$ for $- 54 x + 9 x^{2}$ in the equation $- 16 y^{2} - 54 x + 9 x^{2} = 63$ .

$- 16 y^{2} + 9 {(x - 3)}^{2} - 81 = 63$

Step 1.3

Move $- 81$ to the right side of the equation by adding $81$ to both sides.

$- 16 y^{2} + 9 {(x - 3)}^{2} = 63 + 81$

Step 1.4

Add $63$ and $81$ .

$- 16 y^{2} + 9 {(x - 3)}^{2} = 144$

Step 1.5

Divide each term by $144$ to make the right side equal to one.

$- \frac{16 y^{2}}{144} + \frac{9 {(x - 3)}^{2}}{144} = \frac{144}{144}$

Step 1.6

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 $1$ .

$\frac{{(x - 3)}^{2}}{16} - \frac{y^{2}}{9} = 1$

Step 2

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

$\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$

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, $a$ .

$a = 4$

$b = 3$

$k = 0$

$h = 3$

Step 4

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

$y = \pm \frac{3}{4} \cdot (x - (3)) + 0$

Step 5

Simplify to find the first asymptote.

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Step 5.1

Remove parentheses.

$y = \frac{3}{4} \cdot (x - (3)) + 0$

Step 5.2

Simplify $\frac{3}{4} \cdot (x - (3)) + 0$ .

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Step 5.2.1

Simplify the expression.

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Step 5.2.1.1

Add $\frac{3}{4} \cdot (x - (3))$ and $0$ .

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

Step 5.2.1.2

Multiply $- 1$ by $3$ .

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

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

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

Step 5.2.4

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

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Step 5.2.4.1

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

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

Step 5.2.4.2

Multiply $3$ by $- 3$ .

$y = \frac{3 x}{4} + \frac{- 9}{4}$

Step 5.2.5

Move the negative in front of the fraction.

$y = \frac{3 x}{4} - \frac{9}{4}$

Step 6

Simplify to find the second asymptote.

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Step 6.1

Remove parentheses.

$y = - \frac{3}{4} \cdot (x - (3)) + 0$

Step 6.2

Simplify $- \frac{3}{4} \cdot (x - (3)) + 0$ .

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Step 6.2.1

Simplify the expression.

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Step 6.2.1.1

Add $- \frac{3}{4} \cdot (x - (3))$ and $0$ .

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

Step 6.2.1.2

Multiply $- 1$ by $3$ .

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

Step 6.2.2

Apply the distributive property.

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

Step 6.2.3

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

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

Step 6.2.4

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

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Step 6.2.4.1

Multiply $- 3$ by $- 1$ .

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

Step 6.2.4.2

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

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

Step 6.2.4.3

Multiply $3$ by $3$ .

$y = - \frac{x \cdot 3}{4} + \frac{9}{4}$

Step 6.2.5

Move $3$ to the left of $x$ .

$y = - \frac{3 x}{4} + \frac{9}{4}$

Step 7

This hyperbola has two asymptotes.

$y = \frac{3 x}{4} - \frac{9}{4}, y = - \frac{3 x}{4} + \frac{9}{4}$

Step 8

The asymptotes are $y = \frac{3 x}{4} - \frac{9}{4}$ and $y = - \frac{3 x}{4} + \frac{9}{4}$ .

Asymptotes: $y = \frac{3 x}{4} - \frac{9}{4}, y = - \frac{3 x}{4} + \frac{9}{4}$

Step 9