Precalculus Examples

$16 y^{2} - 25 x^{2} - 96 y + 100 x - 356 = 0$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Add $356$ to both sides of the equation.

$16 y^{2} - 25 x^{2} - 96 y + 100 x = 356$

Step 1.2

Complete the square for $16 y^{2} - 96 y$ .

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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 $c$ .

$a = 16$

$b = - 96$

$c = 0$

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

$d = \frac{- 96}{2 \cdot 16}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 96$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 96$ .

$d = \frac{2 \cdot - 48}{2 \cdot 16}$

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 $2 \cdot 16$ .

$d = \frac{2 \cdot - 48}{2 (16)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 48}{2 \cdot 16}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 48}{16}$

Step 1.2.3.2.2

Cancel the common factor of $- 48$ and $16$ .

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Step 1.2.3.2.2.1

Factor $16$ out of $- 48$ .

$d = \frac{16 \cdot - 3}{16}$

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $16$ out of $16$ .

$d = \frac{16 \cdot - 3}{16 (1)}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{16 \cdot - 3}{16 \cdot 1}$

Step 1.2.3.2.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.2.4

Divide $- 3$ by $1$ .

$d = - 3$

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

$e = 0 - \frac{{(- 96)}^{2}}{4 \cdot 16}$

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 $- 96$ to the power of $2$ .

$e = 0 - \frac{9216}{4 \cdot 16}$

Step 1.2.4.2.1.2

Multiply $4$ by $16$ .

$e = 0 - \frac{9216}{64}$

Step 1.2.4.2.1.3

Divide $9216$ by $64$ .

$e = 0 - 1 \cdot 144$

Step 1.2.4.2.1.4

Multiply $- 1$ by $144$ .

$e = 0 - 144$

Step 1.2.4.2.2

Subtract $144$ from $0$ .

$e = - 144$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(y - 3)}^{2} - 144$ .

$16 {(y - 3)}^{2} - 144$

Step 1.3

Substitute $16 {(y - 3)}^{2} - 144$ for $16 y^{2} - 96 y$ in the equation $16 y^{2} - 25 x^{2} - 96 y + 100 x = 356$ .

$16 {(y - 3)}^{2} - 144 - 25 x^{2} + 100 x = 356$

Step 1.4

Move $- 144$ to the right side of the equation by adding $144$ to both sides.

$16 {(y - 3)}^{2} - 25 x^{2} + 100 x = 356 + 144$

Step 1.5

Complete the square for $- 25 x^{2} + 100 x$ .

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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 $c$ .

$a = - 25$

$b = 100$

$c = 0$

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

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

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Cancel the common factor of $100$ and $2$ .

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Step 1.5.3.2.1.1

Factor $2$ out of $100$ .

$d = \frac{2 \cdot 50}{2 \cdot - 25}$

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 $2 \cdot - 25$ .

$d = \frac{2 \cdot 50}{2 (- 25)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 50}{2 \cdot - 25}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{50}{- 25}$

Step 1.5.3.2.2

Cancel the common factor of $50$ and $- 25$ .

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Step 1.5.3.2.2.1

Factor $25$ out of $50$ .

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

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

$e = 0 - \frac{100^{2}}{4 \cdot - 25}$

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 $100$ to the power of $2$ .

$e = 0 - \frac{10000}{4 \cdot - 25}$

Step 1.5.4.2.1.2

Multiply $4$ by $- 25$ .

$e = 0 - \frac{10000}{- 100}$

Step 1.5.4.2.1.3

Divide $10000$ by $- 100$ .

$e = 0 - - 100$

Step 1.5.4.2.1.4

Multiply $- 1$ by $- 100$ .

$e = 0 + 100$

Step 1.5.4.2.2

Add $0$ and $100$ .

$e = 100$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 25 {(x - 2)}^{2} + 100$ .

$- 25 {(x - 2)}^{2} + 100$

Step 1.6

Substitute $- 25 {(x - 2)}^{2} + 100$ for $- 25 x^{2} + 100 x$ in the equation $16 y^{2} - 25 x^{2} - 96 y + 100 x = 356$ .

$16 {(y - 3)}^{2} - 25 {(x - 2)}^{2} + 100 = 356 + 144$

Step 1.7

Move $100$ to the right side of the equation by adding $100$ to both sides.

$16 {(y - 3)}^{2} - 25 {(x - 2)}^{2} = 356 + 144 - 100$

Step 1.8

Simplify $356 + 144 - 100$ .

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Step 1.8.1

Add $356$ and $144$ .

$16 {(y - 3)}^{2} - 25 {(x - 2)}^{2} = 500 - 100$

Step 1.8.2

Subtract $100$ from $500$ .

$16 {(y - 3)}^{2} - 25 {(x - 2)}^{2} = 400$

Step 1.9

Divide each term by $400$ to make the right side equal to one.

$\frac{16 {(y - 3)}^{2}}{400} - \frac{25 {(x - 2)}^{2}}{400} = \frac{400}{400}$

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 $1$ .

$\frac{{(y - 3)}^{2}}{25} - \frac{{(x - 2)}^{2}}{16} = 1$

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, $a$ .

$a = 5$

$b = 4$

$k = 3$

$h = 2$

Step 4

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

$(2, 3)$

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.

$\sqrt{{(5)}^{2} + {(4)}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Raise $5$ to the power of $2$ .

$\sqrt{25 + {(4)}^{2}}$

Step 5.3.2

Raise $4$ to the power of $2$ .

$\sqrt{25 + 16}$

Step 5.3.3

Add $25$ and $16$ .

$\sqrt{41}$

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.

$(2, 8)$

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.

$(2, - 2)$

Step 6.5

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

$(2, 8), (2, - 2)$

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 $k$ .

$(h, k + c)$

Step 7.2

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

$(2, 3 + \sqrt{41})$

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.

$(2, 3 - \sqrt{41})$

Step 7.5

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

$(2, 3 + \sqrt{41}), (2, 3 - \sqrt{41})$

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.

$\frac{\sqrt{{(5)}^{2} + {(4)}^{2}}}{5}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Raise $5$ to the power of $2$ .

$\frac{\sqrt{25 + 4^{2}}}{5}$

Step 8.3.2

Raise $4$ to the power of $2$ .

$\frac{\sqrt{25 + 16}}{5}$

Step 8.3.3

Add $25$ and $16$ .

$\frac{\sqrt{41}}{5}$

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.

$\frac{4^{2}}{\sqrt{41}}$

Step 9.3

Simplify.

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Step 9.3.1

Raise $4$ to the power of $2$ .

$\frac{16}{\sqrt{41}}$

Step 9.3.2

Multiply $\frac{16}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\frac{16}{\sqrt{41}} \cdot \frac{\sqrt{41}}{\sqrt{41}}$

Step 9.3.3

Combine and simplify the denominator.

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Step 9.3.3.1

Multiply $\frac{16}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\frac{16 \sqrt{41}}{\sqrt{41} \sqrt{41}}$

Step 9.3.3.2

Raise $\sqrt{41}$ to the power of $1$ .

$\frac{16 \sqrt{41}}{{\sqrt{41}}^{1} \sqrt{41}}$

Step 9.3.3.3

Raise $\sqrt{41}$ to the power of $1$ .

$\frac{16 \sqrt{41}}{{\sqrt{41}}^{1} {\sqrt{41}}^{1}}$

Step 9.3.3.4

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

$\frac{16 \sqrt{41}}{{\sqrt{41}}^{1 + 1}}$

Step 9.3.3.5

Add $1$ and $1$ .

$\frac{16 \sqrt{41}}{{\sqrt{41}}^{2}}$

Step 9.3.3.6

Rewrite ${\sqrt{41}}^{2}$ as $41$ .

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Step 9.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{41}$ as $41^{\frac{1}{2}}$ .

$\frac{16 \sqrt{41}}{{(41^{\frac{1}{2}})}^{2}}$

Step 9.3.3.6.2

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

$\frac{16 \sqrt{41}}{41^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

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

$\frac{16 \sqrt{41}}{41^{\frac{2}{2}}}$

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.

$\frac{16 \sqrt{41}}{41^{\frac{2}{2}}}$

Step 9.3.3.6.4.2

Rewrite the expression.

$\frac{16 \sqrt{41}}{41^{1}}$

Step 9.3.3.6.5

Evaluate the exponent.

$\frac{16 \sqrt{41}}{41}$

Step 10

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

$y = \pm \frac{5}{4} \cdot (x - (2)) + 3$

Step 11

Simplify to find the first asymptote.

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Step 11.1

Remove parentheses.

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

Step 11.2

Simplify $\frac{5}{4} \cdot (x - (2)) + 3$ .

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Multiply $- 1$ by $2$ .

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

Step 11.2.1.2

Apply the distributive property.

$y = \frac{5}{4} x + \frac{5}{4} \cdot - 2 + 3$

Step 11.2.1.3

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

$y = \frac{5 x}{4} + \frac{5}{4} \cdot - 2 + 3$

Step 11.2.1.4

Cancel the common factor of $2$ .

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Step 11.2.1.4.1

Factor $2$ out of $4$ .

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

Step 11.2.1.4.2

Factor $2$ out of $- 2$ .

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

Step 11.2.1.4.3

Cancel the common factor.

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

Step 11.2.1.4.4

Rewrite the expression.

$y = \frac{5 x}{4} + \frac{5}{2} \cdot - 1 + 3$

Step 11.2.1.5

Combine $\frac{5}{2}$ and $- 1$ .

$y = \frac{5 x}{4} + \frac{5 \cdot - 1}{2} + 3$

Step 11.2.1.6

Multiply $5$ by $- 1$ .

$y = \frac{5 x}{4} + \frac{- 5}{2} + 3$

Step 11.2.1.7

Move the negative in front of the fraction.

$y = \frac{5 x}{4} - \frac{5}{2} + 3$

Step 11.2.2

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

$y = \frac{5 x}{4} - \frac{5}{2} + 3 \cdot \frac{2}{2}$

Step 11.2.3

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

$y = \frac{5 x}{4} - \frac{5}{2} + \frac{3 \cdot 2}{2}$

Step 11.2.4

Combine the numerators over the common denominator.

$y = \frac{5 x}{4} + \frac{- 5 + 3 \cdot 2}{2}$

Step 11.2.5

Simplify the numerator.

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Step 11.2.5.1

Multiply $3$ by $2$ .

$y = \frac{5 x}{4} + \frac{- 5 + 6}{2}$

Step 11.2.5.2

Add $- 5$ and $6$ .

$y = \frac{5 x}{4} + \frac{1}{2}$

Step 12

Simplify to find the second asymptote.

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Step 12.1

Remove parentheses.

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

Step 12.2

Simplify $- \frac{5}{4} \cdot (x - (2)) + 3$ .

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Multiply $- 1$ by $2$ .

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

Step 12.2.1.2

Apply the distributive property.

$y = - \frac{5}{4} x - \frac{5}{4} \cdot - 2 + 3$

Step 12.2.1.3

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

$y = - \frac{x \cdot 5}{4} - \frac{5}{4} \cdot - 2 + 3$

Step 12.2.1.4

Cancel the common factor of $2$ .

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Step 12.2.1.4.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$y = - \frac{x \cdot 5}{4} + \frac{- 5}{4} \cdot - 2 + 3$

Step 12.2.1.4.2

Factor $2$ out of $4$ .

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

Step 12.2.1.4.3

Factor $2$ out of $- 2$ .

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

Step 12.2.1.4.4

Cancel the common factor.

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

Step 12.2.1.4.5

Rewrite the expression.

$y = - \frac{x \cdot 5}{4} + \frac{- 5}{2} \cdot - 1 + 3$

Step 12.2.1.5

Combine $\frac{- 5}{2}$ and $- 1$ .

$y = - \frac{x \cdot 5}{4} + \frac{- 5 \cdot - 1}{2} + 3$

Step 12.2.1.6

Multiply $- 5$ by $- 1$ .

$y = - \frac{x \cdot 5}{4} + \frac{5}{2} + 3$

Step 12.2.1.7

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

$y = - \frac{5 x}{4} + \frac{5}{2} + 3$

Step 12.2.2

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

$y = - \frac{5 x}{4} + \frac{5}{2} + 3 \cdot \frac{2}{2}$

Step 12.2.3

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

$y = - \frac{5 x}{4} + \frac{5}{2} + \frac{3 \cdot 2}{2}$

Step 12.2.4

Combine the numerators over the common denominator.

$y = - \frac{5 x}{4} + \frac{5 + 3 \cdot 2}{2}$

Step 12.2.5

Simplify the numerator.

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Step 12.2.5.1

Multiply $3$ by $2$ .

$y = - \frac{5 x}{4} + \frac{5 + 6}{2}$

Step 12.2.5.2

Add $5$ and $6$ .

$y = - \frac{5 x}{4} + \frac{11}{2}$

Step 13

This hyperbola has two asymptotes.

$y = \frac{5 x}{4} + \frac{1}{2}, y = - \frac{5 x}{4} + \frac{11}{2}$

Step 14

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

Center: $(2, 3)$

Vertices: $(2, 8), (2, - 2)$

Foci: $(2, 3 + \sqrt{41}), (2, 3 - \sqrt{41})$

Eccentricity: $\frac{\sqrt{41}}{5}$

Focal Parameter: $\frac{16 \sqrt{41}}{41}$

Asymptotes: $y = \frac{5 x}{4} + \frac{1}{2}$ , $y = - \frac{5 x}{4} + \frac{11}{2}$

Step 15