Precalculus Examples

$x^{2} - y^{2} - 4 y = 21$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

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

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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 $c$ .

$a = - 1$

$b = - 4$

$c = 0$

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

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

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

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

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Step 1.1.3.2.1.1

Factor $2$ out of $- 4$ .

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

Step 1.1.3.2.1.2

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

$d = - 1 \cdot - 2$

Step 1.1.3.2.2

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

$d = - - 2$

Step 1.1.3.2.3

Multiply $- 1$ by $- 2$ .

$d = 2$

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

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

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

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

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Step 1.1.4.2.1.1.1

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

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

Step 1.1.4.2.1.1.2

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

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

Step 1.1.4.2.1.1.3

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

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

Step 1.1.4.2.1.1.4

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

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

Step 1.1.4.2.1.1.5

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

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

Step 1.1.4.2.1.1.6

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

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

Step 1.1.4.2.1.2

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

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Step 1.1.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 0 - - 4$

Step 1.1.4.2.1.2.2

Multiply $- 1$ by $- 4$ .

$e = 0 + 4$

Step 1.1.4.2.2

Add $0$ and $4$ .

$e = 4$

Step 1.1.5

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

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

Step 1.2

Substitute $- {(y + 2)}^{2} + 4$ for $- y^{2} - 4 y$ in the equation $x^{2} - y^{2} - 4 y = 21$ .

$x^{2} - {(y + 2)}^{2} + 4 = 21$

Step 1.3

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

$x^{2} - {(y + 2)}^{2} = 21 - 4$

Step 1.4

Subtract $4$ from $21$ .

$x^{2} - {(y + 2)}^{2} = 17$

Step 1.5

Divide each term by $17$ to make the right side equal to one.

$\frac{x^{2}}{17} - \frac{{(y + 2)}^{2}}{17} = \frac{17}{17}$

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 $1$ .

$\frac{x^{2}}{17} - \frac{{(y + 2)}^{2}}{17} = 1$

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.

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

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, $a$ .

$a = \sqrt{17}$

$b = \sqrt{17}$

$k = - 2$

$h = 0$

Step 4

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

$(0, - 2)$

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.

$\sqrt{{(\sqrt{17})}^{2} + {(\sqrt{17})}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 5.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\sqrt{{(17^{\frac{1}{2}})}^{2} + {(\sqrt{17})}^{2}}$

Step 5.3.1.2

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

$\sqrt{17^{\frac{1}{2} \cdot 2} + {(\sqrt{17})}^{2}}$

Step 5.3.1.3

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

$\sqrt{17^{\frac{2}{2}} + {(\sqrt{17})}^{2}}$

Step 5.3.1.4

Cancel the common factor of $2$ .

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Step 5.3.1.4.1

Cancel the common factor.

$\sqrt{17^{\frac{2}{2}} + {(\sqrt{17})}^{2}}$

Step 5.3.1.4.2

Rewrite the expression.

$\sqrt{17^{1} + {(\sqrt{17})}^{2}}$

Step 5.3.1.5

Evaluate the exponent.

$\sqrt{17 + {(\sqrt{17})}^{2}}$

Step 5.3.2

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 5.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\sqrt{17 + {(17^{\frac{1}{2}})}^{2}}$

Step 5.3.2.2

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

$\sqrt{17 + 17^{\frac{1}{2} \cdot 2}}$

Step 5.3.2.3

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

$\sqrt{17 + 17^{\frac{2}{2}}}$

Step 5.3.2.4

Cancel the common factor of $2$ .

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Step 5.3.2.4.1

Cancel the common factor.

$\sqrt{17 + 17^{\frac{2}{2}}}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{17 + 17^{1}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{17 + 17}$

Step 5.3.3

Add $17$ and $17$ .

$\sqrt{34}$

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 $h$ .

$(h + a, k)$

Step 6.2

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

$(\sqrt{17}, - 2)$

Step 6.3

The second vertex of a hyperbola can be found by subtracting $a$ from $h$ .

$(h - a, k)$

Step 6.4

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

$(- \sqrt{17}, - 2)$

Step 6.5

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

$(\sqrt{17}, - 2), (- \sqrt{17}, - 2)$

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 $h$ .

$(h + c, k)$

Step 7.2

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

$(\sqrt{34}, - 2)$

Step 7.3

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

$(h - c, k)$

Step 7.4

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

$(- \sqrt{34}, - 2)$

Step 7.5

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

$(\sqrt{34}, - 2), (- \sqrt{34}, - 2)$

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.

$\frac{\sqrt{{(\sqrt{17})}^{2} + {(\sqrt{17})}^{2}}}{\sqrt{17}}$

Step 8.3

Simplify.

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Step 8.3.1

Simplify the numerator.

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Step 8.3.1.1

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 8.3.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\frac{\sqrt{{(17^{\frac{1}{2}})}^{2} + {\sqrt{17}}^{2}}}{\sqrt{17}}$

Step 8.3.1.1.2

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

$\frac{\sqrt{17^{\frac{1}{2} \cdot 2} + {\sqrt{17}}^{2}}}{\sqrt{17}}$

Step 8.3.1.1.3

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

$\frac{\sqrt{17^{\frac{2}{2}} + {\sqrt{17}}^{2}}}{\sqrt{17}}$

Step 8.3.1.1.4

Cancel the common factor of $2$ .

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Step 8.3.1.1.4.1

Cancel the common factor.

$\frac{\sqrt{17^{\frac{2}{2}} + {\sqrt{17}}^{2}}}{\sqrt{17}}$

Step 8.3.1.1.4.2

Rewrite the expression.

$\frac{\sqrt{17^{1} + {\sqrt{17}}^{2}}}{\sqrt{17}}$

Step 8.3.1.1.5

Evaluate the exponent.

$\frac{\sqrt{17 + {\sqrt{17}}^{2}}}{\sqrt{17}}$

Step 8.3.1.2

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 8.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\frac{\sqrt{17 + {(17^{\frac{1}{2}})}^{2}}}{\sqrt{17}}$

Step 8.3.1.2.2

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

$\frac{\sqrt{17 + 17^{\frac{1}{2} \cdot 2}}}{\sqrt{17}}$

Step 8.3.1.2.3

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

$\frac{\sqrt{17 + 17^{\frac{2}{2}}}}{\sqrt{17}}$

Step 8.3.1.2.4

Cancel the common factor of $2$ .

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Step 8.3.1.2.4.1

Cancel the common factor.

$\frac{\sqrt{17 + 17^{\frac{2}{2}}}}{\sqrt{17}}$

Step 8.3.1.2.4.2

Rewrite the expression.

$\frac{\sqrt{17 + 17^{1}}}{\sqrt{17}}$

Step 8.3.1.2.5

Evaluate the exponent.

$\frac{\sqrt{17 + 17}}{\sqrt{17}}$

Step 8.3.1.3

Add $17$ and $17$ .

$\frac{\sqrt{34}}{\sqrt{17}}$

Step 8.3.2

Combine $\sqrt{34}$ and $\sqrt{17}$ into a single radical.

$\sqrt{\frac{34}{17}}$

Step 8.3.3

Divide $34$ by $17$ .

$\sqrt{2}$

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.

$\frac{{\sqrt{17}}^{2}}{\sqrt{34}}$

Step 9.3

Simplify.

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Step 9.3.1

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 9.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\frac{{(17^{\frac{1}{2}})}^{2}}{\sqrt{34}}$

Step 9.3.1.2

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

$\frac{17^{\frac{1}{2} \cdot 2}}{\sqrt{34}}$

Step 9.3.1.3

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

$\frac{17^{\frac{2}{2}}}{\sqrt{34}}$

Step 9.3.1.4

Cancel the common factor of $2$ .

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Step 9.3.1.4.1

Cancel the common factor.

$\frac{17^{\frac{2}{2}}}{\sqrt{34}}$

Step 9.3.1.4.2

Rewrite the expression.

$\frac{17^{1}}{\sqrt{34}}$

Step 9.3.1.5

Evaluate the exponent.

$\frac{17}{\sqrt{34}}$

Step 9.3.2

Multiply $\frac{17}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\frac{17}{\sqrt{34}} \cdot \frac{\sqrt{34}}{\sqrt{34}}$

Step 9.3.3

Combine and simplify the denominator.

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Step 9.3.3.1

Multiply $\frac{17}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\frac{17 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Step 9.3.3.2

Raise $\sqrt{34}$ to the power of $1$ .

$\frac{17 \sqrt{34}}{{\sqrt{34}}^{1} \sqrt{34}}$

Step 9.3.3.3

Raise $\sqrt{34}$ to the power of $1$ .

$\frac{17 \sqrt{34}}{{\sqrt{34}}^{1} {\sqrt{34}}^{1}}$

Step 9.3.3.4

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

$\frac{17 \sqrt{34}}{{\sqrt{34}}^{1 + 1}}$

Step 9.3.3.5

Add $1$ and $1$ .

$\frac{17 \sqrt{34}}{{\sqrt{34}}^{2}}$

Step 9.3.3.6

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

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Step 9.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\frac{17 \sqrt{34}}{{(34^{\frac{1}{2}})}^{2}}$

Step 9.3.3.6.2

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

$\frac{17 \sqrt{34}}{34^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

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

$\frac{17 \sqrt{34}}{34^{\frac{2}{2}}}$

Step 9.3.3.6.4

Cancel the common factor of $2$ .

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Step 9.3.3.6.4.1

Cancel the common factor.

$\frac{17 \sqrt{34}}{34^{\frac{2}{2}}}$

Step 9.3.3.6.4.2

Rewrite the expression.

$\frac{17 \sqrt{34}}{34^{1}}$

Step 9.3.3.6.5

Evaluate the exponent.

$\frac{17 \sqrt{34}}{34}$

Step 9.3.4

Cancel the common factor of $17$ and $34$ .

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Step 9.3.4.1

Factor $17$ out of $17 \sqrt{34}$ .

$\frac{17 (\sqrt{34})}{34}$

Step 9.3.4.2

Cancel the common factors.

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Step 9.3.4.2.1

Factor $17$ out of $34$ .

$\frac{17 \sqrt{34}}{17 \cdot 2}$

Step 9.3.4.2.2

Cancel the common factor.

$\frac{17 \sqrt{34}}{17 \cdot 2}$

Step 9.3.4.2.3

Rewrite the expression.

$\frac{\sqrt{34}}{2}$

Step 10

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

$y = \pm 1 \cdot x - 2$

Step 11

Multiply $x$ by $1$ .

$y = x - 2$

Step 12

Rewrite $- 1 x$ as $- x$ .

$y = - x - 2$

Step 13

This hyperbola has two asymptotes.

$y = x - 2, y = - x - 2$

Step 14

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

Center: $(0, - 2)$

Vertices: $(\sqrt{17}, - 2), (- \sqrt{17}, - 2)$

Foci: $(\sqrt{34}, - 2), (- \sqrt{34}, - 2)$

Eccentricity: $\sqrt{2}$

Focal Parameter: $\frac{\sqrt{34}}{2}$

Asymptotes: $y = x - 2$ , $y = - x - 2$

Step 15