Precalculus Examples

$x^{2} - 3 y^{2} - 8 x + 12 y + 16 = 0$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Subtract $16$ from both sides of the equation.

$x^{2} - 3 y^{2} - 8 x + 12 y = - 16$

Step 1.2

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

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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 $c$ .

$a = 1$

$b = - 8$

$c = 0$

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

$d = \frac{- 8}{2 \cdot 1}$

Step 1.2.3.2

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

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Step 1.2.3.2.1

Factor $2$ out of $- 8$ .

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

Step 1.2.3.2.2

Cancel the common factors.

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Step 1.2.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

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

Step 1.2.3.2.2.2

Cancel the common factor.

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

Step 1.2.3.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.4

Divide $- 4$ by $1$ .

$d = - 4$

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

$e = 0 - \frac{{(- 8)}^{2}}{4 \cdot 1}$

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

$e = 0 - \frac{64}{4 \cdot 1}$

Step 1.2.4.2.1.2

Multiply $4$ by $1$ .

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

Step 1.2.4.2.1.3

Divide $64$ by $4$ .

$e = 0 - 1 \cdot 16$

Step 1.2.4.2.1.4

Multiply $- 1$ by $16$ .

$e = 0 - 16$

Step 1.2.4.2.2

Subtract $16$ from $0$ .

$e = - 16$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 4)}^{2} - 16$ .

${(x - 4)}^{2} - 16$

Step 1.3

Substitute ${(x - 4)}^{2} - 16$ for $x^{2} - 8 x$ in the equation $x^{2} - 3 y^{2} - 8 x + 12 y = - 16$ .

${(x - 4)}^{2} - 16 - 3 y^{2} + 12 y = - 16$

Step 1.4

Move $- 16$ to the right side of the equation by adding $16$ to both sides.

${(x - 4)}^{2} - 3 y^{2} + 12 y = - 16 + 16$

Step 1.5

Complete the square for $- 3 y^{2} + 12 y$ .

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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 $c$ .

$a = - 3$

$b = 12$

$c = 0$

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

$d = \frac{12}{2 \cdot - 3}$

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Cancel the common factor of $12$ and $2$ .

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Step 1.5.3.2.1.1

Factor $2$ out of $12$ .

$d = \frac{2 \cdot 6}{2 \cdot - 3}$

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 $2 \cdot - 3$ .

$d = \frac{2 \cdot 6}{2 (- 3)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 6}{2 \cdot - 3}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{6}{- 3}$

Step 1.5.3.2.2

Cancel the common factor of $6$ and $- 3$ .

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Step 1.5.3.2.2.1

Factor $3$ out of $6$ .

$d = \frac{3 (2)}{- 3}$

Step 1.5.3.2.2.2

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

$d = - 1 \cdot 2$

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

$e = 0 - \frac{12^{2}}{4 \cdot - 3}$

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 $12$ to the power of $2$ .

$e = 0 - \frac{144}{4 \cdot - 3}$

Step 1.5.4.2.1.2

Multiply $4$ by $- 3$ .

$e = 0 - \frac{144}{- 12}$

Step 1.5.4.2.1.3

Divide $144$ by $- 12$ .

$e = 0 - - 12$

Step 1.5.4.2.1.4

Multiply $- 1$ by $- 12$ .

$e = 0 + 12$

Step 1.5.4.2.2

Add $0$ and $12$ .

$e = 12$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 {(y - 2)}^{2} + 12$ .

$- 3 {(y - 2)}^{2} + 12$

Step 1.6

Substitute $- 3 {(y - 2)}^{2} + 12$ for $- 3 y^{2} + 12 y$ in the equation $x^{2} - 3 y^{2} - 8 x + 12 y = - 16$ .

${(x - 4)}^{2} - 3 {(y - 2)}^{2} + 12 = - 16 + 16$

Step 1.7

Move $12$ to the right side of the equation by adding $12$ to both sides.

${(x - 4)}^{2} - 3 {(y - 2)}^{2} = - 16 + 16 - 12$

Step 1.8

Simplify $- 16 + 16 - 12$ .

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Step 1.8.1

Add $- 16$ and $16$ .

${(x - 4)}^{2} - 3 {(y - 2)}^{2} = 0 - 12$

Step 1.8.2

Subtract $12$ from $0$ .

${(x - 4)}^{2} - 3 {(y - 2)}^{2} = - 12$

Step 1.9

Flip the sign on each term of the equation so the term on the right side is positive.

$- {(x - 4)}^{2} + 3 {(y - 2)}^{2} = 12$

Step 1.10

Divide each term by $12$ to make the right side equal to one.

$- \frac{{(x - 4)}^{2}}{12} + \frac{3 {(y - 2)}^{2}}{12} = \frac{12}{12}$

Step 1.11

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 $1$ .

$\frac{{(y - 2)}^{2}}{4} - \frac{{(x - 4)}^{2}}{12} = 1$

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, $a$ .

$a = 2$

$b = 2 \sqrt{3}$

$k = 2$

$h = 4$

Step 4

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

$(4, 2)$

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.

$\sqrt{{(2)}^{2} + {(2 \sqrt{3})}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the expression.

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Step 5.3.1.1

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

$\sqrt{4 + {(2 \sqrt{3})}^{2}}$

Step 5.3.1.2

Apply the product rule to $2 \sqrt{3}$ .

$\sqrt{4 + 2^{2} {\sqrt{3}}^{2}}$

Step 5.3.1.3

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

$\sqrt{4 + 4 {\sqrt{3}}^{2}}$

Step 5.3.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{4 + 4 {(3^{\frac{1}{2}})}^{2}}$

Step 5.3.2.2

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

$\sqrt{4 + 4 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 5.3.2.3

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

$\sqrt{4 + 4 \cdot 3^{\frac{2}{2}}}$

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.

$\sqrt{4 + 4 \cdot 3^{\frac{2}{2}}}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{4 + 4 \cdot 3^{1}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{4 + 4 \cdot 3}$

Step 5.3.3

Simplify the expression.

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Step 5.3.3.1

Multiply $4$ by $3$ .

$\sqrt{4 + 12}$

Step 5.3.3.2

Add $4$ and $12$ .

$\sqrt{16}$

Step 5.3.3.3

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2}}$

Step 5.3.3.4

Pull terms out from under the radical, assuming positive real numbers.

$4$

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.

$(4, 4)$

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.

$(4, 0)$

Step 6.5

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

$(4, 4), (4, 0)$

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.

$(4, 6)$

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.

$(4, - 2)$

Step 7.5

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

$(4, 6), (4, - 2)$

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.

$\frac{\sqrt{{(2)}^{2} + {(2 \sqrt{3})}^{2}}}{2}$

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 $2$ .

$\frac{\sqrt{4 + {(2 \sqrt{3})}^{2}}}{2}$

Step 8.3.1.2

Apply the product rule to $2 \sqrt{3}$ .

$\frac{\sqrt{4 + 2^{2} {\sqrt{3}}^{2}}}{2}$

Step 8.3.1.3

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

$\frac{\sqrt{4 + 4 {\sqrt{3}}^{2}}}{2}$

Step 8.3.1.4

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{4 + 4 {(3^{\frac{1}{2}})}^{2}}}{2}$

Step 8.3.1.4.2

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

$\frac{\sqrt{4 + 4 \cdot 3^{\frac{1}{2} \cdot 2}}}{2}$

Step 8.3.1.4.3

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

$\frac{\sqrt{4 + 4 \cdot 3^{\frac{2}{2}}}}{2}$

Step 8.3.1.4.4

Cancel the common factor of $2$ .

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Step 8.3.1.4.4.1

Cancel the common factor.

$\frac{\sqrt{4 + 4 \cdot 3^{\frac{2}{2}}}}{2}$

Step 8.3.1.4.4.2

Rewrite the expression.

$\frac{\sqrt{4 + 4 \cdot 3^{1}}}{2}$

Step 8.3.1.4.5

Evaluate the exponent.

$\frac{\sqrt{4 + 4 \cdot 3}}{2}$

Step 8.3.1.5

Multiply $4$ by $3$ .

$\frac{\sqrt{4 + 12}}{2}$

Step 8.3.1.6

Add $4$ and $12$ .

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

Step 8.3.1.7

Rewrite $16$ as $4^{2}$ .

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

Step 8.3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$\frac{4}{2}$

Step 8.3.2

Divide $4$ by $2$ .

$2$

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.

$\frac{2 {\sqrt{3}}^{2}}{4}$

Step 9.3

Simplify.

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Step 9.3.1

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

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Step 9.3.1.1

Factor $2$ out of $2 {\sqrt{3}}^{2}$ .

$\frac{2 ({\sqrt{3}}^{2})}{4}$

Step 9.3.1.2

Cancel the common factors.

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Step 9.3.1.2.1

Factor $2$ out of $4$ .

$\frac{2 {\sqrt{3}}^{2}}{2 \cdot 2}$

Step 9.3.1.2.2

Cancel the common factor.

$\frac{2 {\sqrt{3}}^{2}}{2 \cdot 2}$

Step 9.3.1.2.3

Rewrite the expression.

$\frac{{\sqrt{3}}^{2}}{2}$

Step 9.3.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 9.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{{(3^{\frac{1}{2}})}^{2}}{2}$

Step 9.3.2.2

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

$\frac{3^{\frac{1}{2} \cdot 2}}{2}$

Step 9.3.2.3

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

$\frac{3^{\frac{2}{2}}}{2}$

Step 9.3.2.4

Cancel the common factor of $2$ .

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Step 9.3.2.4.1

Cancel the common factor.

$\frac{3^{\frac{2}{2}}}{2}$

Step 9.3.2.4.2

Rewrite the expression.

$\frac{3^{1}}{2}$

Step 9.3.2.5

Evaluate the exponent.

$\frac{3}{2}$

Step 10

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

$y = \pm \frac{\sqrt{3}}{3} \cdot (x - (4)) + 2$

Step 11

Simplify to find the first asymptote.

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Step 11.1

Remove parentheses.

$y = \frac{\sqrt{3}}{3} \cdot (x - (4)) + 2$

Step 11.2

Simplify each term.

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Step 11.2.1

Multiply $- 1$ by $4$ .

$y = \frac{\sqrt{3}}{3} \cdot (x - 4) + 2$

Step 11.2.2

Apply the distributive property.

$y = \frac{\sqrt{3}}{3} x + \frac{\sqrt{3}}{3} \cdot - 4 + 2$

Step 11.2.3

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

$y = \frac{\sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \cdot - 4 + 2$

Step 11.2.4

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

$y = \frac{\sqrt{3} x}{3} + \frac{\sqrt{3} \cdot - 4}{3} + 2$

Step 11.2.5

Simplify each term.

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Step 11.2.5.1

Move $- 4$ to the left of $\sqrt{3}$ .

$y = \frac{\sqrt{3} x}{3} + \frac{- 4 \sqrt{3}}{3} + 2$

Step 11.2.5.2

Move the negative in front of the fraction.

$y = \frac{\sqrt{3} x}{3} - \frac{4 \sqrt{3}}{3} + 2$

Step 12

Simplify to find the second asymptote.

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Step 12.1

Remove parentheses.

$y = - \frac{\sqrt{3}}{3} \cdot (x - (4)) + 2$

Step 12.2

Simplify each term.

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Step 12.2.1

Multiply $- 1$ by $4$ .

$y = - \frac{\sqrt{3}}{3} \cdot (x - 4) + 2$

Step 12.2.2

Apply the distributive property.

$y = - \frac{\sqrt{3}}{3} x - \frac{\sqrt{3}}{3} \cdot - 4 + 2$

Step 12.2.3

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

$y = - \frac{x \sqrt{3}}{3} - \frac{\sqrt{3}}{3} \cdot - 4 + 2$

Step 12.2.4

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

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Step 12.2.4.1

Multiply $- 4$ by $- 1$ .

$y = - \frac{x \sqrt{3}}{3} + 4 \frac{\sqrt{3}}{3} + 2$

Step 12.2.4.2

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

$y = - \frac{x \sqrt{3}}{3} + \frac{4 \sqrt{3}}{3} + 2$

Step 13

This hyperbola has two asymptotes.

$y = \frac{\sqrt{3} x}{3} - \frac{4 \sqrt{3}}{3} + 2, y = - \frac{x \sqrt{3}}{3} + \frac{4 \sqrt{3}}{3} + 2$

Step 14

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

Center: $(4, 2)$

Vertices: $(4, 4), (4, 0)$

Foci: $(4, 6), (4, - 2)$

Eccentricity: $2$

Focal Parameter: $\frac{3}{2}$

Asymptotes: $y = \frac{\sqrt{3} x}{3} - \frac{4 \sqrt{3}}{3} + 2$ , $y = - \frac{x \sqrt{3}}{3} + \frac{4 \sqrt{3}}{3} + 2$

Step 15