Precalculus Examples

$9 x^{2} - 4 y^{2} - 90 x - 24 y = - 153$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

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

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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 $c$ .

$a = 9$

$b = - 90$

$c = 0$

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

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

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

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

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Step 1.1.3.2.1.1

Factor $2$ out of $- 90$ .

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

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 $2 \cdot 9$ .

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

Step 1.1.3.2.1.2.2

Cancel the common factor.

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

Step 1.1.3.2.1.2.3

Rewrite the expression.

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

Step 1.1.3.2.2

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

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Step 1.1.3.2.2.1

Factor $9$ out of $- 45$ .

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

Step 1.1.3.2.2.2

Cancel the common factors.

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Step 1.1.3.2.2.2.1

Factor $9$ out of $9$ .

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

Step 1.1.3.2.2.2.2

Cancel the common factor.

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

Step 1.1.3.2.2.2.3

Rewrite the expression.

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

Step 1.1.3.2.2.2.4

Divide $- 5$ by $1$ .

$d = - 5$

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

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

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

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

Step 1.1.4.2.1.2

Multiply $4$ by $9$ .

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

Step 1.1.4.2.1.3

Divide $8100$ by $36$ .

$e = 0 - 1 \cdot 225$

Step 1.1.4.2.1.4

Multiply $- 1$ by $225$ .

$e = 0 - 225$

Step 1.1.4.2.2

Subtract $225$ from $0$ .

$e = - 225$

Step 1.1.5

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

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

Step 1.2

Substitute $9 {(x - 5)}^{2} - 225$ for $9 x^{2} - 90 x$ in the equation $9 x^{2} - 4 y^{2} - 90 x - 24 y = - 153$ .

$9 {(x - 5)}^{2} - 225 - 4 y^{2} - 24 y = - 153$

Step 1.3

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

$9 {(x - 5)}^{2} - 4 y^{2} - 24 y = - 153 + 225$

Step 1.4

Complete the square for $- 4 y^{2} - 24 y$ .

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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 $c$ .

$a = - 4$

$b = - 24$

$c = 0$

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

$d = \frac{- 24}{2 \cdot - 4}$

Step 1.4.3.2

Simplify the right side.

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Step 1.4.3.2.1

Cancel the common factor of $- 24$ and $2$ .

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Step 1.4.3.2.1.1

Factor $2$ out of $- 24$ .

$d = \frac{2 \cdot - 12}{2 \cdot - 4}$

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 $2 \cdot - 4$ .

$d = \frac{2 \cdot - 12}{2 (- 4)}$

Step 1.4.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 12}{2 \cdot - 4}$

Step 1.4.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 12}{- 4}$

Step 1.4.3.2.2

Cancel the common factor of $- 12$ and $- 4$ .

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Step 1.4.3.2.2.1

Factor $- 4$ out of $- 12$ .

$d = \frac{- 4 (3)}{- 4}$

Step 1.4.3.2.2.2

Cancel the common factors.

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Step 1.4.3.2.2.2.1

Factor $- 4$ out of $- 4$ .

$d = \frac{- 4 \cdot 3}{- 4 \cdot 1}$

Step 1.4.3.2.2.2.2

Cancel the common factor.

$d = \frac{- 4 \cdot 3}{- 4 \cdot 1}$

Step 1.4.3.2.2.2.3

Rewrite the expression.

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

Step 1.4.3.2.2.2.4

Divide $3$ by $1$ .

$d = 3$

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

$e = 0 - \frac{{(- 24)}^{2}}{4 \cdot - 4}$

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 $- 24$ to the power of $2$ .

$e = 0 - \frac{576}{4 \cdot - 4}$

Step 1.4.4.2.1.2

Multiply $4$ by $- 4$ .

$e = 0 - \frac{576}{- 16}$

Step 1.4.4.2.1.3

Divide $576$ by $- 16$ .

$e = 0 - - 36$

Step 1.4.4.2.1.4

Multiply $- 1$ by $- 36$ .

$e = 0 + 36$

Step 1.4.4.2.2

Add $0$ and $36$ .

$e = 36$

Step 1.4.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(y + 3)}^{2} + 36$ .

$- 4 {(y + 3)}^{2} + 36$

Step 1.5

Substitute $- 4 {(y + 3)}^{2} + 36$ for $- 4 y^{2} - 24 y$ in the equation $9 x^{2} - 4 y^{2} - 90 x - 24 y = - 153$ .

$9 {(x - 5)}^{2} - 4 {(y + 3)}^{2} + 36 = - 153 + 225$

Step 1.6

Move $36$ to the right side of the equation by adding $36$ to both sides.

$9 {(x - 5)}^{2} - 4 {(y + 3)}^{2} = - 153 + 225 - 36$

Step 1.7

Simplify $- 153 + 225 - 36$ .

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Step 1.7.1

Add $- 153$ and $225$ .

$9 {(x - 5)}^{2} - 4 {(y + 3)}^{2} = 72 - 36$

Step 1.7.2

Subtract $36$ from $72$ .

$9 {(x - 5)}^{2} - 4 {(y + 3)}^{2} = 36$

Step 1.8

Divide each term by $36$ to make the right side equal to one.

$\frac{9 {(x - 5)}^{2}}{36} - \frac{4 {(y + 3)}^{2}}{36} = \frac{36}{36}$

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 $1$ .

$\frac{{(x - 5)}^{2}}{4} - \frac{{(y + 3)}^{2}}{9} = 1$

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, $a$ .

$a = 2$

$b = 3$

$k = - 3$

$h = 5$

Step 4

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

$(5, - 3)$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

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

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

Step 5.3.2

Raise $3$ to the power of $2$ .

$\sqrt{4 + 9}$

Step 5.3.3

Add $4$ and $9$ .

$\sqrt{13}$

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.

$(7, - 3)$

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.

$(3, - 3)$

Step 6.5

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

$(7, - 3), (3, - 3)$

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.

$(5 + \sqrt{13}, - 3)$

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.

$(5 - \sqrt{13}, - 3)$

Step 7.5

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

$(5 + \sqrt{13}, - 3), (5 - \sqrt{13}, - 3)$

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.

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

Step 8.3

Simplify the numerator.

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Step 8.3.1

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

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

Step 8.3.2

Raise $3$ to the power of $2$ .

$\frac{\sqrt{4 + 9}}{2}$

Step 8.3.3

Add $4$ and $9$ .

$\frac{\sqrt{13}}{2}$

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.

$\frac{3^{2}}{\sqrt{13}}$

Step 9.3

Simplify.

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Step 9.3.1

Raise $3$ to the power of $2$ .

$\frac{9}{\sqrt{13}}$

Step 9.3.2

Multiply $\frac{9}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\frac{9}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}$

Step 9.3.3

Combine and simplify the denominator.

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Step 9.3.3.1

Multiply $\frac{9}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\frac{9 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Step 9.3.3.2

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{9 \sqrt{13}}{{\sqrt{13}}^{1} \sqrt{13}}$

Step 9.3.3.3

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{9 \sqrt{13}}{{\sqrt{13}}^{1} {\sqrt{13}}^{1}}$

Step 9.3.3.4

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

$\frac{9 \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

Step 9.3.3.5

Add $1$ and $1$ .

$\frac{9 \sqrt{13}}{{\sqrt{13}}^{2}}$

Step 9.3.3.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 9.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{9 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

Step 9.3.3.6.2

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

$\frac{9 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

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

$\frac{9 \sqrt{13}}{13^{\frac{2}{2}}}$

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.

$\frac{9 \sqrt{13}}{13^{\frac{2}{2}}}$

Step 9.3.3.6.4.2

Rewrite the expression.

$\frac{9 \sqrt{13}}{13^{1}}$

Step 9.3.3.6.5

Evaluate the exponent.

$\frac{9 \sqrt{13}}{13}$

Step 10

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

$y = \pm \frac{3}{2} \cdot (x - (5)) - 3$

Step 11

Simplify to find the first asymptote.

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Step 11.1

Remove parentheses.

$y = \frac{3}{2} \cdot (x - (5)) - 3$

Step 11.2

Simplify $\frac{3}{2} \cdot (x - (5)) - 3$ .

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Multiply $- 1$ by $5$ .

$y = \frac{3}{2} \cdot (x - 5) - 3$

Step 11.2.1.2

Apply the distributive property.

$y = \frac{3}{2} x + \frac{3}{2} \cdot - 5 - 3$

Step 11.2.1.3

Combine $\frac{3}{2}$ and $x$ .

$y = \frac{3 x}{2} + \frac{3}{2} \cdot - 5 - 3$

Step 11.2.1.4

Multiply $\frac{3}{2} \cdot - 5$ .

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Step 11.2.1.4.1

Combine $\frac{3}{2}$ and $- 5$ .

$y = \frac{3 x}{2} + \frac{3 \cdot - 5}{2} - 3$

Step 11.2.1.4.2

Multiply $3$ by $- 5$ .

$y = \frac{3 x}{2} + \frac{- 15}{2} - 3$

Step 11.2.1.5

Move the negative in front of the fraction.

$y = \frac{3 x}{2} - \frac{15}{2} - 3$

Step 11.2.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{3 x}{2} - \frac{15}{2} - 3 \cdot \frac{2}{2}$

Step 11.2.3

Combine $- 3$ and $\frac{2}{2}$ .

$y = \frac{3 x}{2} - \frac{15}{2} + \frac{- 3 \cdot 2}{2}$

Step 11.2.4

Combine the numerators over the common denominator.

$y = \frac{3 x}{2} + \frac{- 15 - 3 \cdot 2}{2}$

Step 11.2.5

Simplify the numerator.

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Step 11.2.5.1

Multiply $- 3$ by $2$ .

$y = \frac{3 x}{2} + \frac{- 15 - 6}{2}$

Step 11.2.5.2

Subtract $6$ from $- 15$ .

$y = \frac{3 x}{2} + \frac{- 21}{2}$

Step 11.2.6

Move the negative in front of the fraction.

$y = \frac{3 x}{2} - \frac{21}{2}$

Step 12

Simplify to find the second asymptote.

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Step 12.1

Remove parentheses.

$y = - \frac{3}{2} \cdot (x - (5)) - 3$

Step 12.2

Simplify $- \frac{3}{2} \cdot (x - (5)) - 3$ .

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Multiply $- 1$ by $5$ .

$y = - \frac{3}{2} \cdot (x - 5) - 3$

Step 12.2.1.2

Apply the distributive property.

$y = - \frac{3}{2} x - \frac{3}{2} \cdot - 5 - 3$

Step 12.2.1.3

Combine $x$ and $\frac{3}{2}$ .

$y = - \frac{x \cdot 3}{2} - \frac{3}{2} \cdot - 5 - 3$

Step 12.2.1.4

Multiply $- \frac{3}{2} \cdot - 5$ .

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Step 12.2.1.4.1

Multiply $- 5$ by $- 1$ .

$y = - \frac{x \cdot 3}{2} + 5 (\frac{3}{2}) - 3$

Step 12.2.1.4.2

Combine $5$ and $\frac{3}{2}$ .

$y = - \frac{x \cdot 3}{2} + \frac{5 \cdot 3}{2} - 3$

Step 12.2.1.4.3

Multiply $5$ by $3$ .

$y = - \frac{x \cdot 3}{2} + \frac{15}{2} - 3$

Step 12.2.1.5

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

$y = - \frac{3 x}{2} + \frac{15}{2} - 3$

Step 12.2.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{3 x}{2} + \frac{15}{2} - 3 \cdot \frac{2}{2}$

Step 12.2.3

Combine $- 3$ and $\frac{2}{2}$ .

$y = - \frac{3 x}{2} + \frac{15}{2} + \frac{- 3 \cdot 2}{2}$

Step 12.2.4

Combine the numerators over the common denominator.

$y = - \frac{3 x}{2} + \frac{15 - 3 \cdot 2}{2}$

Step 12.2.5

Simplify the numerator.

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Step 12.2.5.1

Multiply $- 3$ by $2$ .

$y = - \frac{3 x}{2} + \frac{15 - 6}{2}$

Step 12.2.5.2

Subtract $6$ from $15$ .

$y = - \frac{3 x}{2} + \frac{9}{2}$

Step 13

This hyperbola has two asymptotes.

$y = \frac{3 x}{2} - \frac{21}{2}, y = - \frac{3 x}{2} + \frac{9}{2}$

Step 14

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

Center: $(5, - 3)$

Vertices: $(7, - 3), (3, - 3)$

Foci: $(5 + \sqrt{13}, - 3), (5 - \sqrt{13}, - 3)$

Eccentricity: $\frac{\sqrt{13}}{2}$

Focal Parameter: $\frac{9 \sqrt{13}}{13}$

Asymptotes: $y = \frac{3 x}{2} - \frac{21}{2}$ , $y = - \frac{3 x}{2} + \frac{9}{2}$

Step 15