Precalculus Examples

$- 4 x^{2} + 40 x + 25 y^{2} - 100 y + 100 = 0$

Step 1

Find the standard form of the hyperbola.

Tap for more steps...

Step 1.1

Subtract $100$ from both sides of the equation.

$- 4 x^{2} + 40 x + 25 y^{2} - 100 y = - 100$

Step 1.2

Complete the square for $- 4 x^{2} + 40 x$ .

Tap for more steps...

Step 1.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 4$

$b = 40$

$c = 0$

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

Tap for more steps...

Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{40}{2 \cdot - 4}$

Step 1.2.3.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $40$ and $2$ .

Tap for more steps...

Step 1.2.3.2.1.1

Factor $2$ out of $40$ .

$d = \frac{2 \cdot 20}{2 \cdot - 4}$

Step 1.2.3.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.2.1.2.1

Factor $2$ out of $2 \cdot - 4$ .

$d = \frac{2 \cdot 20}{2 (- 4)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 20}{2 \cdot - 4}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{20}{- 4}$

Step 1.2.3.2.2

Cancel the common factor of $20$ and $- 4$ .

Tap for more steps...

Step 1.2.3.2.2.1

Factor $4$ out of $20$ .

$d = \frac{4 (5)}{- 4}$

Step 1.2.3.2.2.2

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$d = - 1 \cdot 5$

Step 1.2.3.2.3

Multiply $- 1$ by $5$ .

$d = - 5$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Tap for more steps...

Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{40^{2}}{4 \cdot - 4}$

Step 1.2.4.2

Simplify the right side.

Tap for more steps...

Step 1.2.4.2.1

Simplify each term.

Tap for more steps...

Step 1.2.4.2.1.1

Raise $40$ to the power of $2$ .

$e = 0 - \frac{1600}{4 \cdot - 4}$

Step 1.2.4.2.1.2

Multiply $4$ by $- 4$ .

$e = 0 - \frac{1600}{- 16}$

Step 1.2.4.2.1.3

Divide $1600$ by $- 16$ .

$e = 0 - - 100$

Step 1.2.4.2.1.4

Multiply $- 1$ by $- 100$ .

$e = 0 + 100$

Step 1.2.4.2.2

Add $0$ and $100$ .

$e = 100$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(x - 5)}^{2} + 100$ .

$- 4 {(x - 5)}^{2} + 100$

Step 1.3

Substitute $- 4 {(x - 5)}^{2} + 100$ for $- 4 x^{2} + 40 x$ in the equation $- 4 x^{2} + 40 x + 25 y^{2} - 100 y = - 100$ .

$- 4 {(x - 5)}^{2} + 100 + 25 y^{2} - 100 y = - 100$

Step 1.4

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

$- 4 {(x - 5)}^{2} + 25 y^{2} - 100 y = - 100 - 100$

Step 1.5

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

Tap for more steps...

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

Tap for more steps...

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.

Tap for more steps...

Step 1.5.3.2.1

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

Tap for more steps...

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.

Tap for more steps...

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 $25$ .

Tap for more steps...

Step 1.5.3.2.2.1

Factor $25$ out of $- 50$ .

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

Step 1.5.3.2.2.2

Cancel the common factors.

Tap for more steps...

Step 1.5.3.2.2.2.1

Factor $25$ out of $25$ .

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

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

Tap for more steps...

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.

Tap for more steps...

Step 1.5.4.2.1

Simplify each term.

Tap for more steps...

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 $100$ .

$e = 0 - 1 \cdot 100$

Step 1.5.4.2.1.4

Multiply $- 1$ by $100$ .

$e = 0 - 100$

Step 1.5.4.2.2

Subtract $100$ from $0$ .

$e = - 100$

Step 1.5.5

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

$25 {(y - 2)}^{2} - 100$

Step 1.6

Substitute $25 {(y - 2)}^{2} - 100$ for $25 y^{2} - 100 y$ in the equation $- 4 x^{2} + 40 x + 25 y^{2} - 100 y = - 100$ .

$- 4 {(x - 5)}^{2} + 25 {(y - 2)}^{2} - 100 = - 100 - 100$

Step 1.7

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

$- 4 {(x - 5)}^{2} + 25 {(y - 2)}^{2} = - 100 - 100 + 100$

Step 1.8

Simplify $- 100 - 100 + 100$ .

Tap for more steps...

Step 1.8.1

Subtract $100$ from $- 100$ .

$- 4 {(x - 5)}^{2} + 25 {(y - 2)}^{2} = - 200 + 100$

Step 1.8.2

Add $- 200$ and $100$ .

$- 4 {(x - 5)}^{2} + 25 {(y - 2)}^{2} = - 100$

Step 1.9

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

$4 {(x - 5)}^{2} - 25 {(y - 2)}^{2} = 100$

Step 1.10

Divide each term by $100$ to make the right side equal to one.

$\frac{4 {(x - 5)}^{2}}{100} - \frac{25 {(y - 2)}^{2}}{100} = \frac{100}{100}$

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 $1$ .

$\frac{{(x - 5)}^{2}}{25} - \frac{{(y - 2)}^{2}}{4} = 1$

Step 2

This is the form of a hyperbola. Use this form to determine the values used to find the 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, $a$ .

$a = 5$

$b = 2$

$k = 2$

$h = 5$

Step 4

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

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

Step 5

Simplify to find the first asymptote.

Tap for more steps...

Step 5.1

Remove parentheses.

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Multiply $- 1$ by $5$ .

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

Step 5.2.1.2

Apply the distributive property.

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

Step 5.2.1.3

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

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

Step 5.2.1.4

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.1.4.1

Factor $5$ out of $- 5$ .

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

Step 5.2.1.4.2

Cancel the common factor.

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

Step 5.2.1.4.3

Rewrite the expression.

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

Step 5.2.1.5

Multiply $2$ by $- 1$ .

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

Step 5.2.2

Combine the opposite terms in $\frac{2 x}{5} - 2 + 2$ .

Tap for more steps...

Step 5.2.2.1

Add $- 2$ and $2$ .

$y = \frac{2 x}{5} + 0$

Step 5.2.2.2

Add $\frac{2 x}{5}$ and $0$ .

$y = \frac{2 x}{5}$

Step 6

Simplify to find the second asymptote.

Tap for more steps...

Step 6.1

Remove parentheses.

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

Step 6.2

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

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Multiply $- 1$ by $5$ .

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

Step 6.2.1.2

Apply the distributive property.

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

Step 6.2.1.3

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

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

Step 6.2.1.4

Cancel the common factor of $5$ .

Tap for more steps...

Step 6.2.1.4.1

Move the leading negative in $- \frac{2}{5}$ into the numerator.

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

Step 6.2.1.4.2

Factor $5$ out of $- 5$ .

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

Step 6.2.1.4.3

Cancel the common factor.

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

Step 6.2.1.4.4

Rewrite the expression.

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

Step 6.2.1.5

Multiply $- 2$ by $- 1$ .

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

Step 6.2.1.6

Move $2$ to the left of $x$ .

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

Step 6.2.2

Add $2$ and $2$ .

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

Step 7

This hyperbola has two asymptotes.

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

Step 8

The asymptotes are $y = \frac{2 x}{5}$ and $y = - \frac{2 x}{5} + 4$ .

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

Step 9