Precalculus Examples

$11 x^{2} - 25 y^{2} + 22 x + 250 y - 889 = 0$

Step 1

Find the standard form of the hyperbola.

Tap for more steps...

Step 1.1

Add $889$ to both sides of the equation.

$11 x^{2} - 25 y^{2} + 22 x + 250 y = 889$

Step 1.2

Complete the square for $11 x^{2} + 22 x$ .

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 $c$ .

$a = 11$

$b = 22$

$c = 0$

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

$d = \frac{22}{2 \cdot 11}$

Step 1.2.3.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $22$ and $2$ .

Tap for more steps...

Step 1.2.3.2.1.1

Factor $2$ out of $22$ .

$d = \frac{2 \cdot 11}{2 \cdot 11}$

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 $2 \cdot 11$ .

$d = \frac{2 \cdot 11}{2 (11)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 11}{2 \cdot 11}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{11}{11}$

Step 1.2.3.2.2

Cancel the common factor of $11$ .

Tap for more steps...

Step 1.2.3.2.2.1

Cancel the common factor.

$d = \frac{11}{11}$

Step 1.2.3.2.2.2

Rewrite the expression.

$d = 1$

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

$e = 0 - \frac{22^{2}}{4 \cdot 11}$

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 $22$ to the power of $2$ .

$e = 0 - \frac{484}{4 \cdot 11}$

Step 1.2.4.2.1.2

Multiply $4$ by $11$ .

$e = 0 - \frac{484}{44}$

Step 1.2.4.2.1.3

Divide $484$ by $44$ .

$e = 0 - 1 \cdot 11$

Step 1.2.4.2.1.4

Multiply $- 1$ by $11$ .

$e = 0 - 11$

Step 1.2.4.2.2

Subtract $11$ from $0$ .

$e = - 11$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $11 {(x + 1)}^{2} - 11$ .

$11 {(x + 1)}^{2} - 11$

Step 1.3

Substitute $11 {(x + 1)}^{2} - 11$ for $11 x^{2} + 22 x$ in the equation $11 x^{2} - 25 y^{2} + 22 x + 250 y = 889$ .

$11 {(x + 1)}^{2} - 11 - 25 y^{2} + 250 y = 889$

Step 1.4

Move $- 11$ to the right side of the equation by adding $11$ to both sides.

$11 {(x + 1)}^{2} - 25 y^{2} + 250 y = 889 + 11$

Step 1.5

Complete the square for $- 25 y^{2} + 250 y$ .

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 $c$ .

$a = - 25$

$b = 250$

$c = 0$

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

$d = \frac{250}{2 \cdot - 25}$

Step 1.5.3.2

Simplify the right side.

Tap for more steps...

Step 1.5.3.2.1

Cancel the common factor of $250$ and $2$ .

Tap for more steps...

Step 1.5.3.2.1.1

Factor $2$ out of $250$ .

$d = \frac{2 \cdot 125}{2 \cdot - 25}$

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 $2 \cdot - 25$ .

$d = \frac{2 \cdot 125}{2 (- 25)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 125}{2 \cdot - 25}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{125}{- 25}$

Step 1.5.3.2.2

Cancel the common factor of $125$ and $- 25$ .

Tap for more steps...

Step 1.5.3.2.2.1

Factor $25$ out of $125$ .

$d = \frac{25 (5)}{- 25}$

Step 1.5.3.2.2.2

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

$d = - 1 \cdot 5$

Step 1.5.3.2.3

Multiply $- 1$ by $5$ .

$d = - 5$

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

$e = 0 - \frac{250^{2}}{4 \cdot - 25}$

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 $250$ to the power of $2$ .

$e = 0 - \frac{62500}{4 \cdot - 25}$

Step 1.5.4.2.1.2

Multiply $4$ by $- 25$ .

$e = 0 - \frac{62500}{- 100}$

Step 1.5.4.2.1.3

Divide $62500$ by $- 100$ .

$e = 0 - - 625$

Step 1.5.4.2.1.4

Multiply $- 1$ by $- 625$ .

$e = 0 + 625$

Step 1.5.4.2.2

Add $0$ and $625$ .

$e = 625$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 25 {(y - 5)}^{2} + 625$ .

$- 25 {(y - 5)}^{2} + 625$

Step 1.6

Substitute $- 25 {(y - 5)}^{2} + 625$ for $- 25 y^{2} + 250 y$ in the equation $11 x^{2} - 25 y^{2} + 22 x + 250 y = 889$ .

$11 {(x + 1)}^{2} - 25 {(y - 5)}^{2} + 625 = 889 + 11$

Step 1.7

Move $625$ to the right side of the equation by adding $625$ to both sides.

$11 {(x + 1)}^{2} - 25 {(y - 5)}^{2} = 889 + 11 - 625$

Step 1.8

Simplify $889 + 11 - 625$ .

Tap for more steps...

Step 1.8.1

Add $889$ and $11$ .

$11 {(x + 1)}^{2} - 25 {(y - 5)}^{2} = 900 - 625$

Step 1.8.2

Subtract $625$ from $900$ .

$11 {(x + 1)}^{2} - 25 {(y - 5)}^{2} = 275$

Step 1.9

Divide each term by $275$ to make the right side equal to one.

$\frac{11 {(x + 1)}^{2}}{275} - \frac{25 {(y - 5)}^{2}}{275} = \frac{275}{275}$

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 $1$ .

$\frac{{(x + 1)}^{2}}{25} - \frac{{(y - 5)}^{2}}{11} = 1$

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, $a$ .

$a = 5$

$b = \sqrt{11}$

$k = 5$

$h = - 1$

Step 4

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

$(- 1, 5)$

Step 5

Find $c$ , the distance from the center to a focus.

Tap for more steps...

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.

$\sqrt{{(5)}^{2} + {(\sqrt{11})}^{2}}$

Step 5.3

Simplify.

Tap for more steps...

Step 5.3.1

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

$\sqrt{25 + {(\sqrt{11})}^{2}}$

Step 5.3.2

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Tap for more steps...

Step 5.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\sqrt{25 + {(11^{\frac{1}{2}})}^{2}}$

Step 5.3.2.2

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

$\sqrt{25 + 11^{\frac{1}{2} \cdot 2}}$

Step 5.3.2.3

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

$\sqrt{25 + 11^{\frac{2}{2}}}$

Step 5.3.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.3.2.4.1

Cancel the common factor.

$\sqrt{25 + 11^{\frac{2}{2}}}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{25 + 11^{1}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{25 + 11}$

Step 5.3.3

Simplify the expression.

Tap for more steps...

Step 5.3.3.1

Add $25$ and $11$ .

$\sqrt{36}$

Step 5.3.3.2

Rewrite $36$ as $6^{2}$ .

$\sqrt{6^{2}}$

Step 5.3.3.3

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

$6$

Step 6

Find the vertices.

Tap for more steps...

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.

$(4, 5)$

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.

$(- 6, 5)$

Step 6.5

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

$(4, 5), (- 6, 5)$

Step 7

Find the foci.

Tap for more steps...

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.

$(5, 5)$

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.

$(- 7, 5)$

Step 7.5

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

$(5, 5), (- 7, 5)$

Step 8

Find the eccentricity.

Tap for more steps...

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.

$\frac{\sqrt{{(5)}^{2} + {(\sqrt{11})}^{2}}}{5}$

Step 8.3

Simplify the numerator.

Tap for more steps...

Step 8.3.1

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

$\frac{\sqrt{25 + {\sqrt{11}}^{2}}}{5}$

Step 8.3.2

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Tap for more steps...

Step 8.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{\sqrt{25 + {(11^{\frac{1}{2}})}^{2}}}{5}$

Step 8.3.2.2

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

$\frac{\sqrt{25 + 11^{\frac{1}{2} \cdot 2}}}{5}$

Step 8.3.2.3

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

$\frac{\sqrt{25 + 11^{\frac{2}{2}}}}{5}$

Step 8.3.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.3.2.4.1

Cancel the common factor.

$\frac{\sqrt{25 + 11^{\frac{2}{2}}}}{5}$

Step 8.3.2.4.2

Rewrite the expression.

$\frac{\sqrt{25 + 11^{1}}}{5}$

Step 8.3.2.5

Evaluate the exponent.

$\frac{\sqrt{25 + 11}}{5}$

Step 8.3.3

Add $25$ and $11$ .

$\frac{\sqrt{36}}{5}$

Step 8.3.4

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{6^{2}}}{5}$

Step 8.3.5

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

$\frac{6}{5}$

Step 9

Find the focal parameter.

Tap for more steps...

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.

$\frac{{\sqrt{11}}^{2}}{6}$

Step 9.3

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Tap for more steps...

Step 9.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{{(11^{\frac{1}{2}})}^{2}}{6}$

Step 9.3.2

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

$\frac{11^{\frac{1}{2} \cdot 2}}{6}$

Step 9.3.3

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

$\frac{11^{\frac{2}{2}}}{6}$

Step 9.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.3.4.1

Cancel the common factor.

$\frac{11^{\frac{2}{2}}}{6}$

Step 9.3.4.2

Rewrite the expression.

$\frac{11^{1}}{6}$

Step 9.3.5

Evaluate the exponent.

$\frac{11}{6}$

Step 10

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

$y = \pm \frac{\sqrt{11}}{5} \cdot (x - (- 1)) + 5$

Step 11

Simplify to find the first asymptote.

Tap for more steps...

Step 11.1

Remove parentheses.

$y = \frac{\sqrt{11}}{5} \cdot (x - (- 1)) + 5$

Step 11.2

Simplify each term.

Tap for more steps...

Step 11.2.1

Multiply $- 1$ by $- 1$ .

$y = \frac{\sqrt{11}}{5} \cdot (x + 1) + 5$

Step 11.2.2

Apply the distributive property.

$y = \frac{\sqrt{11}}{5} x + \frac{\sqrt{11}}{5} \cdot 1 + 5$

Step 11.2.3

Combine $\frac{\sqrt{11}}{5}$ and $x$ .

$y = \frac{\sqrt{11} x}{5} + \frac{\sqrt{11}}{5} \cdot 1 + 5$

Step 11.2.4

Multiply $\frac{\sqrt{11}}{5}$ by $1$ .

$y = \frac{\sqrt{11} x}{5} + \frac{\sqrt{11}}{5} + 5$

Step 12

Simplify to find the second asymptote.

Tap for more steps...

Step 12.1

Remove parentheses.

$y = - \frac{\sqrt{11}}{5} \cdot (x - (- 1)) + 5$

Step 12.2

Simplify each term.

Tap for more steps...

Step 12.2.1

Multiply $- 1$ by $- 1$ .

$y = - \frac{\sqrt{11}}{5} \cdot (x + 1) + 5$

Step 12.2.2

Apply the distributive property.

$y = - \frac{\sqrt{11}}{5} x - \frac{\sqrt{11}}{5} \cdot 1 + 5$

Step 12.2.3

Combine $x$ and $\frac{\sqrt{11}}{5}$ .

$y = - \frac{x \sqrt{11}}{5} - \frac{\sqrt{11}}{5} \cdot 1 + 5$

Step 12.2.4

Multiply $- 1$ by $1$ .

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

Step 13

This hyperbola has two asymptotes.

$y = \frac{\sqrt{11} x}{5} + \frac{\sqrt{11}}{5} + 5, y = - \frac{x \sqrt{11}}{5} - \frac{\sqrt{11}}{5} + 5$

Step 14

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

Center: $(- 1, 5)$

Vertices: $(4, 5), (- 6, 5)$

Foci: $(5, 5), (- 7, 5)$

Eccentricity: $\frac{6}{5}$

Focal Parameter: $\frac{11}{6}$

Asymptotes: $y = \frac{\sqrt{11} x}{5} + \frac{\sqrt{11}}{5} + 5$ , $y = - \frac{x \sqrt{11}}{5} - \frac{\sqrt{11}}{5} + 5$

Step 15