Find the Hyperbola: Center (0,0), Focus (0,2), Vertex (0,1) (0,0) , (0,2) , (0,1)

$(0, 0)$ , $(0, 2)$ , $(0, 1)$

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

$a$ is the distance between the vertex $(0, 1)$ and the center point $(0, 0)$ .

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

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

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

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

Simplify.

Subtract $0$ from $0$ .

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

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

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

Subtract $0$ from $1$ .

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

One to any power is one.

$a = \sqrt{0 + 1}$

Add $0$ and $1$ .

$a = \sqrt{1}$

Any root of $1$ is $1$ .

$a = 1$

$c$ is the distance between the focus $(0, 2)$ and the center $(0, 0)$ .

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

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

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

$c = \sqrt{{(0 - 0)}^{2} + {(2 - 0)}^{2}}$

Simplify.

Subtract $0$ from $0$ .

$c = \sqrt{0^{2} + {(2 - 0)}^{2}}$

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

$c = \sqrt{0 + {(2 - 0)}^{2}}$

Subtract $0$ from $2$ .

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

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

$c = \sqrt{0 + 4}$

Add $0$ and $4$ .

$c = \sqrt{4}$

Rewrite $4$ as $2^{2}$ .

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

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

$c = 2$

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

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

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

One to any power is one.

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

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

$1 + b^{2} = 4$

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

Subtract $1$ from both sides of the equation.

$b^{2} = 4 - 1$

Subtract $1$ from $4$ .

$b^{2} = 3$

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

$b = \pm \sqrt{3}$

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

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

$b = \sqrt{3}$

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

$b = - \sqrt{3}$

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

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

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

$b = \sqrt{3}$

The slope of the line between the focus $(0, 2)$ and the center $(0, 0)$ 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.

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

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

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).

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

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

$m = \frac{0 - (2)}{0 - (0)}$

Simplify.

Subtract $0$ from $0$ .

$\frac{0 - (2)}{0}$

The expression contains a division by $0$ . The expression is undefined.

Undefined

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

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

Substitute the values $h = 0$ , $k = 0$ , $a = 1$ , and $b = \sqrt{3}$ into $\frac{{(y - k)}^{2}}{a^{2}} - \frac{{(x - h)}^{2}}{b^{2}} = 1$ to get the hyperbola equation $\frac{{(y - (0))}^{2}}{{(1)}^{2}} - \frac{{(x - (0))}^{2}}{{(\sqrt{3})}^{2}} = 1$ .

$\frac{{(y - (0))}^{2}}{{(1)}^{2}} - \frac{{(x - (0))}^{2}}{{(\sqrt{3})}^{2}} = 1$

Simplify to find the final equation of the hyperbola.

Simplify the numerator.

Multiply $- 1$ by $0$ .

$\frac{{(y + 0)}^{2}}{1^{2}} - \frac{{(x - (0))}^{2}}{{(\sqrt{3})}^{2}} = 1$

Add $y$ and $0$ .

$\frac{y^{2}}{1^{2}} - \frac{{(x - (0))}^{2}}{{(\sqrt{3})}^{2}} = 1$

Simplify the expression.

One to any power is one.

$\frac{y^{2}}{1} - \frac{{(x - (0))}^{2}}{{(\sqrt{3})}^{2}} = 1$

Divide $y^{2}$ by $1$ .

$y^{2} - \frac{{(x - (0))}^{2}}{{(\sqrt{3})}^{2}} = 1$

Simplify the numerator.

Multiply $- 1$ by $0$ .

$y^{2} - \frac{{(x + 0)}^{2}}{{\sqrt{3}}^{2}} = 1$

Add $x$ and $0$ .

$y^{2} - \frac{x^{2}}{{\sqrt{3}}^{2}} = 1$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

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

$y^{2} - \frac{x^{2}}{3^{\frac{1}{2} \cdot 2}} = 1$

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

$y^{2} - \frac{x^{2}}{3^{\frac{2}{2}}} = 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$y^{2} - \frac{x^{2}}{3^{\frac{2}{2}}} = 1$

Rewrite the expression.

$y^{2} - \frac{x^{2}}{3} = 1$

Evaluate the exponent.

$y^{2} - \frac{x^{2}}{3} = 1$

Find the Hyperbola: Center (0,0), Focus (0,2), Vertex (0,1) (0,0) , (0,2) , (0,1)

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