Graph (x^2)/10-(y^2)/30=1

$\frac{x^{2}}{10} - \frac{y^{2}}{30} = 1$

Step 1

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 $1$ .

$\frac{x^{2}}{10} - \frac{y^{2}}{30} = 1$

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, $a$ .

$a = \sqrt{10}$

$b = \sqrt{30}$

$k = 0$

$h = 0$

Step 4

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

$(0, 0)$

Step 5

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

Find the distance from the center to a focus of the hyperbola by using the following formula.

$\sqrt{a^{2} + b^{2}}$

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(\sqrt{10})}^{2} + {(\sqrt{30})}^{2}}$

Simplify.

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sqrt{{(10^{\frac{1}{2}})}^{2} + {(\sqrt{30})}^{2}}$

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

$\sqrt{10^{\frac{1}{2} \cdot 2} + {(\sqrt{30})}^{2}}$

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

$\sqrt{10^{\frac{2}{2}} + {(\sqrt{30})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{10^{\frac{2}{2}} + {(\sqrt{30})}^{2}}$

Rewrite the expression.

$\sqrt{10^{1} + {(\sqrt{30})}^{2}}$

Evaluate the exponent.

$\sqrt{10 + {(\sqrt{30})}^{2}}$

Rewrite ${\sqrt{30}}^{2}$ as $30$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{30}$ as $30^{\frac{1}{2}}$ .

$\sqrt{10 + {(30^{\frac{1}{2}})}^{2}}$

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

$\sqrt{10 + 30^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{10 + 30^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{10 + 30^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{10 + 30^{1}}$

Evaluate the exponent.

$\sqrt{10 + 30}$

Add $10$ and $30$ .

$\sqrt{40}$

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$\sqrt{4 (10)}$

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

$\sqrt{2^{2} \cdot 10}$

Pull terms out from under the radical.

$2 \sqrt{10}$

Step 6

Find the vertices.

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

$(h + a, k)$

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

$(\sqrt{10}, 0)$

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

$(h - a, k)$

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

$(- \sqrt{10}, 0)$

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

$(\sqrt{10}, 0), (- \sqrt{10}, 0)$

Step 7

Find the foci.

The first focus of a hyperbola can be found by adding $c$ to $h$ .

$(h + c, k)$

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

$(2 \sqrt{10}, 0)$

The second focus of a hyperbola can be found by subtracting $c$ from $h$ .

$(h - c, k)$

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

$(- 2 \sqrt{10}, 0)$

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

$(2 \sqrt{10}, 0), (- 2 \sqrt{10}, 0)$

Step 8

Find the eccentricity.

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} + b^{2}}}{a}$

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(\sqrt{10})}^{2} + {(\sqrt{30})}^{2}}}{\sqrt{10}}$

Simplify.

Simplify the numerator.

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{\sqrt{{(10^{\frac{1}{2}})}^{2} + {\sqrt{30}}^{2}}}{\sqrt{10}}$

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

$\frac{\sqrt{10^{\frac{1}{2} \cdot 2} + {\sqrt{30}}^{2}}}{\sqrt{10}}$

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

$\frac{\sqrt{10^{\frac{2}{2}} + {\sqrt{30}}^{2}}}{\sqrt{10}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{10^{\frac{2}{2}} + {\sqrt{30}}^{2}}}{\sqrt{10}}$

Rewrite the expression.

$\frac{\sqrt{10^{1} + {\sqrt{30}}^{2}}}{\sqrt{10}}$

Evaluate the exponent.

$\frac{\sqrt{10 + {\sqrt{30}}^{2}}}{\sqrt{10}}$

Rewrite ${\sqrt{30}}^{2}$ as $30$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{30}$ as $30^{\frac{1}{2}}$ .

$\frac{\sqrt{10 + {(30^{\frac{1}{2}})}^{2}}}{\sqrt{10}}$

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

$\frac{\sqrt{10 + 30^{\frac{1}{2} \cdot 2}}}{\sqrt{10}}$

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

$\frac{\sqrt{10 + 30^{\frac{2}{2}}}}{\sqrt{10}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{10 + 30^{\frac{2}{2}}}}{\sqrt{10}}$

Rewrite the expression.

$\frac{\sqrt{10 + 30^{1}}}{\sqrt{10}}$

Evaluate the exponent.

$\frac{\sqrt{10 + 30}}{\sqrt{10}}$

Add $10$ and $30$ .

$\frac{\sqrt{40}}{\sqrt{10}}$

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$\frac{\sqrt{4 (10)}}{\sqrt{10}}$

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

$\frac{\sqrt{2^{2} \cdot 10}}{\sqrt{10}}$

Pull terms out from under the radical.

$\frac{2 \sqrt{10}}{\sqrt{10}}$

Cancel the common factor of $\sqrt{10}$ .

Cancel the common factor.

$\frac{2 \sqrt{10}}{\sqrt{10}}$

Divide $2$ by $1$ .

$2$

Step 9

Find the focal parameter.

Find the value of the focal parameter of the hyperbola by using the following formula.

$\frac{b^{2}}{\sqrt{a^{2} + b^{2}}}$

Substitute the values of $b$ and $\sqrt{a^{2} + b^{2}}$ in the formula.

$\frac{{\sqrt{30}}^{2}}{2 \sqrt{10}}$

Simplify.

Rewrite ${\sqrt{30}}^{2}$ as $30$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{30}$ as $30^{\frac{1}{2}}$ .

$\frac{{(30^{\frac{1}{2}})}^{2}}{2 \sqrt{10}}$

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

$\frac{30^{\frac{1}{2} \cdot 2}}{2 \sqrt{10}}$

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

$\frac{30^{\frac{2}{2}}}{2 \sqrt{10}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{30^{\frac{2}{2}}}{2 \sqrt{10}}$

Rewrite the expression.

$\frac{30^{1}}{2 \sqrt{10}}$

Evaluate the exponent.

$\frac{30}{2 \sqrt{10}}$

Cancel the common factor of $30$ and $2$ .

Factor $2$ out of $30$ .

$\frac{2 \cdot 15}{2 \sqrt{10}}$

Cancel the common factors.

Factor $2$ out of $2 \sqrt{10}$ .

$\frac{2 \cdot 15}{2 (\sqrt{10})}$

Cancel the common factor.

$\frac{2 \cdot 15}{2 \sqrt{10}}$

Rewrite the expression.

$\frac{15}{\sqrt{10}}$

Multiply $\frac{15}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{15}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Combine and simplify the denominator.

Multiply $\frac{15}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{15 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{15 \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{15 \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{15 \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{15 \sqrt{10}}{{\sqrt{10}}^{2}}$

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{15 \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

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

$\frac{15 \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

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

$\frac{15 \sqrt{10}}{10^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{15 \sqrt{10}}{10^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{15 \sqrt{10}}{10^{1}}$

Evaluate the exponent.

$\frac{15 \sqrt{10}}{10}$

Cancel the common factor of $15$ and $10$ .

Factor $5$ out of $15 \sqrt{10}$ .

$\frac{5 (3 \sqrt{10})}{10}$

Cancel the common factors.

Factor $5$ out of $10$ .

$\frac{5 (3 \sqrt{10})}{5 (2)}$

Cancel the common factor.

$\frac{5 (3 \sqrt{10})}{5 \cdot 2}$

Rewrite the expression.

$\frac{3 \sqrt{10}}{2}$

Step 10

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

$y = \pm \sqrt{3} \cdot x + 0$

Step 11

Add $\sqrt{3} \cdot x$ and $0$ .

$y = \sqrt{3} x$

Step 12

Add $- \sqrt{3} \cdot x$ and $0$ .

$y = - \sqrt{3} x$

Step 13

This hyperbola has two asymptotes.

$y = \sqrt{3} x, y = - \sqrt{3} x$

Step 14

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

Center: $(0, 0)$

Vertices: $(\sqrt{10}, 0), (- \sqrt{10}, 0)$

Foci: $(2 \sqrt{10}, 0), (- 2 \sqrt{10}, 0)$

Eccentricity: $2$

Focal Parameter: $\frac{3 \sqrt{10}}{2}$

Asymptotes: $y = \sqrt{3} x$ , $y = - \sqrt{3} x$

Step 15

Graph (x^2)/10-(y^2)/30=1

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