Examples

$(5, 1)$ , $(4, 1)$ , $(- 5, 1)$

Step 1

There are two general equations for a hyperbola.

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

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

Step 2

$a$ is the distance between the vertex $(4, 1)$ and the center point $(5, 1)$ .

Tap for more steps...

Step 2.1

Use the distance formula to determine the distance between the two points.

$Distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Step 2.2

Substitute the actual values of the points into the distance formula.

$a = \sqrt{{(4 - 5)}^{2} + {(1 - 1)}^{2}}$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

Subtract $5$ from $4$ .

$a = \sqrt{{(- 1)}^{2} + {(1 - 1)}^{2}}$

Step 2.3.2

Raise $- 1$ to the power of $2$ .

$a = \sqrt{1 + {(1 - 1)}^{2}}$

Step 2.3.3

Subtract $1$ from $1$ .

$a = \sqrt{1 + 0^{2}}$

Step 2.3.4

Raising $0$ to any positive power yields $0$ .

$a = \sqrt{1 + 0}$

Step 2.3.5

Add $1$ and $0$ .

$a = \sqrt{1}$

Step 2.3.6

Any root of $1$ is $1$ .

$a = 1$

Step 3

$c$ is the distance between the focus $(- 5, 1)$ and the center $(5, 1)$ .

Tap for more steps...

Step 3.1

Use the distance formula to determine the distance between the two points.

$Distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Step 3.2

Substitute the actual values of the points into the distance formula.

$c = \sqrt{{((- 5) - 5)}^{2} + {(1 - 1)}^{2}}$

Step 3.3

Simplify.

Tap for more steps...

Step 3.3.1

Subtract $5$ from $- 5$ .

$c = \sqrt{{(- 10)}^{2} + {(1 - 1)}^{2}}$

Step 3.3.2

Raise $- 10$ to the power of $2$ .

$c = \sqrt{100 + {(1 - 1)}^{2}}$

Step 3.3.3

Subtract $1$ from $1$ .

$c = \sqrt{100 + 0^{2}}$

Step 3.3.4

Raising $0$ to any positive power yields $0$ .

$c = \sqrt{100 + 0}$

Step 3.3.5

Add $100$ and $0$ .

$c = \sqrt{100}$

Step 3.3.6

Rewrite $100$ as $10^{2}$ .

$c = \sqrt{10^{2}}$

Step 3.3.7

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

$c = 10$

Step 4

Using the equation $c^{2} = a^{2} + b^{2}$ . Substitute $1$ for $a$ and $10$ for $c$ .

Tap for more steps...

Step 4.1

Rewrite the equation as ${(1)}^{2} + b^{2} = 10^{2}$ .

${(1)}^{2} + b^{2} = 10^{2}$

Step 4.2

One to any power is one.

$1 + b^{2} = 10^{2}$

Step 4.3

Raise $10$ to the power of $2$ .

$1 + b^{2} = 100$

Step 4.4

Move all terms not containing $b$ to the right side of the equation.

Tap for more steps...

Step 4.4.1

Subtract $1$ from both sides of the equation.

$b^{2} = 100 - 1$

Step 4.4.2

Subtract $1$ from $100$ .

$b^{2} = 99$

Step 4.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$b = \pm \sqrt{99}$

Step 4.6

Simplify $\pm \sqrt{99}$ .

Tap for more steps...

Step 4.6.1

Rewrite $99$ as $3^{2} \cdot 11$ .

Tap for more steps...

Step 4.6.1.1

Factor $9$ out of $99$ .

$b = \pm \sqrt{9 (11)}$

Step 4.6.1.2

Rewrite $9$ as $3^{2}$ .

$b = \pm \sqrt{3^{2} \cdot 11}$

Step 4.6.2

Pull terms out from under the radical.

$b = \pm 3 \sqrt{11}$

Step 4.7

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 4.7.1

First, use the positive value of the $\pm$ to find the first solution.

$b = 3 \sqrt{11}$

Step 4.7.2

Next, use the negative value of the $\pm$ to find the second solution.

$b = - 3 \sqrt{11}$

Step 4.7.3

The complete solution is the result of both the positive and negative portions of the solution.

$b = 3 \sqrt{11}, - 3 \sqrt{11}$

Step 5

$b$ is a distance, which means it should be a positive number.

$b = 3 \sqrt{11}$

Step 6

The slope of the line between the focus $(- 5, 1)$ and the center $(5, 1)$ determines whether the hyperbola is vertical or horizontal. If the slope is $0$ , the graph is horizontal. If the slope is undefined, the graph is vertical.

Tap for more steps...

Step 6.1

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 6.2

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 6.3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{1 - (1)}{5 - (- 5)}$

Step 6.4

Simplify.

Tap for more steps...

Step 6.4.1

Simplify the numerator.

Tap for more steps...

Step 6.4.1.1

Multiply $- 1$ by $1$ .

$m = \frac{1 - 1}{5 - (- 5)}$

Step 6.4.1.2

Subtract $1$ from $1$ .

$m = \frac{0}{5 - (- 5)}$

Step 6.4.2

Simplify the denominator.

Tap for more steps...

Step 6.4.2.1

Multiply $- 1$ by $- 5$ .

$m = \frac{0}{5 + 5}$

Step 6.4.2.2

Add $5$ and $5$ .

$m = \frac{0}{10}$

Step 6.4.3

Divide $0$ by $10$ .

$m = 0$

Step 6.5

The general equation for a horizontal hyperbola is $\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$ .

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

Step 7

Substitute the values $h = 5$ , $k = 1$ , $a = 1$ , and $b = 3 \sqrt{11}$ into $\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$ to get the hyperbola equation $\frac{{(x - (5))}^{2}}{{(1)}^{2}} - \frac{{(y - (1))}^{2}}{{(3 \sqrt{11})}^{2}} = 1$ .

$\frac{{(x - (5))}^{2}}{{(1)}^{2}} - \frac{{(y - (1))}^{2}}{{(3 \sqrt{11})}^{2}} = 1$

Step 8

Simplify to find the final equation of the hyperbola.

Tap for more steps...

Step 8.1

Multiply $- 1$ by $5$ .

$\frac{{(x - 5)}^{2}}{1^{2}} - \frac{{(y - (1))}^{2}}{{(3 \sqrt{11})}^{2}} = 1$

Step 8.2

One to any power is one.

$\frac{{(x - 5)}^{2}}{1} - \frac{{(y - (1))}^{2}}{{(3 \sqrt{11})}^{2}} = 1$

Step 8.3

Divide ${(x - 5)}^{2}$ by $1$ .

${(x - 5)}^{2} - \frac{{(y - (1))}^{2}}{{(3 \sqrt{11})}^{2}} = 1$

Step 8.4

Multiply $- 1$ by $1$ .

${(x - 5)}^{2} - \frac{{(y - 1)}^{2}}{{(3 \sqrt{11})}^{2}} = 1$

Step 8.5

Simplify the denominator.

Tap for more steps...

Step 8.5.1

Apply the product rule to $3 \sqrt{11}$ .

${(x - 5)}^{2} - \frac{{(y - 1)}^{2}}{3^{2} {\sqrt{11}}^{2}} = 1$

Step 8.5.2

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

${(x - 5)}^{2} - \frac{{(y - 1)}^{2}}{9 {\sqrt{11}}^{2}} = 1$

Step 8.5.3

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

Tap for more steps...

Step 8.5.3.1

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

${(x - 5)}^{2} - \frac{{(y - 1)}^{2}}{9 {(11^{\frac{1}{2}})}^{2}} = 1$

Step 8.5.3.2

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

${(x - 5)}^{2} - \frac{{(y - 1)}^{2}}{9 \cdot 11^{\frac{1}{2} \cdot 2}} = 1$

Step 8.5.3.3

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

${(x - 5)}^{2} - \frac{{(y - 1)}^{2}}{9 \cdot 11^{\frac{2}{2}}} = 1$

Step 8.5.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.5.3.4.1

Cancel the common factor.

${(x - 5)}^{2} - \frac{{(y - 1)}^{2}}{9 \cdot 11^{\frac{2}{2}}} = 1$

Step 8.5.3.4.2

Rewrite the expression.

${(x - 5)}^{2} - \frac{{(y - 1)}^{2}}{9 \cdot 11} = 1$

Step 8.5.3.5

Evaluate the exponent.

${(x - 5)}^{2} - \frac{{(y - 1)}^{2}}{9 \cdot 11} = 1$

Step 8.6

Multiply $9$ by $11$ .

${(x - 5)}^{2} - \frac{{(y - 1)}^{2}}{99} = 1$

Step 9

Enter YOUR Problem