Algebra Examples

$9 y^{2} - 25 x^{2} - 100 x - 108 y - 1 = 0$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Add $1$ to both sides of the equation.

$9 y^{2} - 25 x^{2} - 100 x - 108 y = 1$

Step 1.2

Complete the square for $9 y^{2} - 108 y$ .

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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 $c$ .

$a = 9$

$b = - 108$

$c = 0$

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

$d = \frac{- 108}{2 \cdot 9}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 108$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 108$ .

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

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 $2 \cdot 9$ .

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

Step 1.2.3.2.1.2.2

Cancel the common factor.

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

Step 1.2.3.2.1.2.3

Rewrite the expression.

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

Step 1.2.3.2.2

Cancel the common factor of $- 54$ and $9$ .

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Step 1.2.3.2.2.1

Factor $9$ out of $- 54$ .

$d = \frac{9 \cdot - 6}{9}$

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 6}{9 (1)}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot - 6}{9 \cdot 1}$

Step 1.2.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 6}{1}$

Step 1.2.3.2.2.2.4

Divide $- 6$ by $1$ .

$d = - 6$

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

$e = 0 - \frac{{(- 108)}^{2}}{4 \cdot 9}$

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 $- 108$ to the power of $2$ .

$e = 0 - \frac{11664}{4 \cdot 9}$

Step 1.2.4.2.1.2

Multiply $4$ by $9$ .

$e = 0 - \frac{11664}{36}$

Step 1.2.4.2.1.3

Divide $11664$ by $36$ .

$e = 0 - 1 \cdot 324$

Step 1.2.4.2.1.4

Multiply $- 1$ by $324$ .

$e = 0 - 324$

Step 1.2.4.2.2

Subtract $324$ from $0$ .

$e = - 324$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(y - 6)}^{2} - 324$ .

$9 {(y - 6)}^{2} - 324$

Step 1.3

Substitute $9 {(y - 6)}^{2} - 324$ for $9 y^{2} - 108 y$ in the equation $9 y^{2} - 25 x^{2} - 100 x - 108 y = 1$ .

$9 {(y - 6)}^{2} - 324 - 25 x^{2} - 100 x = 1$

Step 1.4

Move $- 324$ to the right side of the equation by adding $324$ to both sides.

$9 {(y - 6)}^{2} - 25 x^{2} - 100 x = 1 + 324$

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

Cancel the common factors.

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Step 1.5.3.2.2.2.1

Factor $- 25$ out of $- 25$ .

$d = \frac{- 25 \cdot 2}{- 25 \cdot 1}$

Step 1.5.3.2.2.2.2

Cancel the common factor.

$d = \frac{- 25 \cdot 2}{- 25 \cdot 1}$

Step 1.5.3.2.2.2.3

Rewrite the expression.

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

Step 1.5.3.2.2.2.4

Divide $2$ by $1$ .

$d = 2$

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 $9 y^{2} - 25 x^{2} - 100 x - 108 y = 1$ .

$9 {(y - 6)}^{2} - 25 {(x + 2)}^{2} + 100 = 1 + 324$

Step 1.7

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

$9 {(y - 6)}^{2} - 25 {(x + 2)}^{2} = 1 + 324 - 100$

Step 1.8

Simplify $1 + 324 - 100$ .

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Step 1.8.1

Add $1$ and $324$ .

$9 {(y - 6)}^{2} - 25 {(x + 2)}^{2} = 325 - 100$

Step 1.8.2

Subtract $100$ from $325$ .

$9 {(y - 6)}^{2} - 25 {(x + 2)}^{2} = 225$

Step 1.9

Divide each term by $225$ to make the right side equal to one.

$\frac{9 {(y - 6)}^{2}}{225} - \frac{25 {(x + 2)}^{2}}{225} = \frac{225}{225}$

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 $1$ .

$\frac{{(y - 6)}^{2}}{25} - \frac{{(x + 2)}^{2}}{9} = 1$

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.

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

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, $a$ .

$a = 5$

$b = 3$

$k = 6$

$h = - 2$

Step 4

Find the vertices.

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Step 4.1

The first vertex of a hyperbola can be found by adding $a$ to $k$ .

$(h, k + a)$

Step 4.2

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

$(- 2, 11)$

Step 4.3

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

$(h, k - a)$

Step 4.4

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

$(- 2, 1)$

Step 4.5

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

$(- 2, 11), (- 2, 1)$

Step 5