Find the Axis of Symmetry 5x^2-4y^2+2x-1=0

$5 x^{2} - 4 y^{2} + 2 x - 1 = 0$

Step 1

Find the standard form of the hyperbola.

Add $1$ to both sides of the equation.

$5 x^{2} - 4 y^{2} + 2 x = 1$

Complete the square for $5 x^{2} + 2 x$ .

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

$a = 5$

$b = 2$

$c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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

$d = \frac{2}{2 \cdot 5}$

Cancel the common factor of $2$ .

Cancel the common factor.

$d = \frac{2}{2 \cdot 5}$

Rewrite the expression.

$d = \frac{1}{5}$

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

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

$e = 0 - \frac{2^{2}}{4 \cdot 5}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{4}{4 \cdot 5}$

Multiply $4$ by $5$ .

$e = 0 - \frac{4}{20}$

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

Factor $4$ out of $4$ .

$e = 0 - \frac{4 (1)}{20}$

Cancel the common factors.

Factor $4$ out of $20$ .

$e = 0 - \frac{4 \cdot 1}{4 \cdot 5}$

Cancel the common factor.

$e = 0 - \frac{4 \cdot 1}{4 \cdot 5}$

Rewrite the expression.

$e = 0 - \frac{1}{5}$

Subtract $\frac{1}{5}$ from $0$ .

$e = - \frac{1}{5}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $5 {(x + \frac{1}{5})}^{2} - \frac{1}{5}$ .

$5 {(x + \frac{1}{5})}^{2} - \frac{1}{5}$

Substitute $5 {(x + \frac{1}{5})}^{2} - \frac{1}{5}$ for $5 x^{2} + 2 x$ in the equation $5 x^{2} - 4 y^{2} + 2 x = 1$ .

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

Move $- \frac{1}{5}$ to the right side of the equation by adding $\frac{1}{5}$ to both sides.

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

Simplify $1 + \frac{1}{5}$ .

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $5$ and $1$ .

$5 {(x + \frac{1}{5})}^{2} - 4 y^{2} = \frac{6}{5}$

Divide each term by $\frac{6}{5}$ to make the right side equal to one.

$\frac{5 {(x + \frac{1}{5})}^{2}}{\frac{6}{5}} - \frac{4 y^{2}}{\frac{6}{5}} = \frac{\frac{6}{5}}{\frac{6}{5}}$

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 $1$ .

$\frac{{(x + \frac{1}{5})}^{2}}{\frac{6}{25}} - \frac{y^{2}}{\frac{3}{10}} = 1$

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, $a$ .

$a = \frac{\sqrt{6}}{5}$

$b = \frac{\sqrt{30}}{10}$

$k = 0$

$h = - \frac{1}{5}$

Step 4

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

$(- \frac{1}{5}, 0)$

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.

$\sqrt{{(\frac{\sqrt{6}}{5})}^{2} + {(\frac{\sqrt{30}}{10})}^{2}}$

Simplify.

Apply the product rule to $\frac{\sqrt{6}}{5}$ .

$\sqrt{\frac{{\sqrt{6}}^{2}}{5^{2}} + {(\frac{\sqrt{30}}{10})}^{2}}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sqrt{\frac{{(6^{\frac{1}{2}})}^{2}}{5^{2}} + {(\frac{\sqrt{30}}{10})}^{2}}$

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

$\sqrt{\frac{6^{\frac{1}{2} \cdot 2}}{5^{2}} + {(\frac{\sqrt{30}}{10})}^{2}}$

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

$\sqrt{\frac{6^{\frac{2}{2}}}{5^{2}} + {(\frac{\sqrt{30}}{10})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{\frac{6^{\frac{2}{2}}}{5^{2}} + {(\frac{\sqrt{30}}{10})}^{2}}$

Rewrite the expression.

$\sqrt{\frac{6^{1}}{5^{2}} + {(\frac{\sqrt{30}}{10})}^{2}}$

Evaluate the exponent.

$\sqrt{\frac{6}{5^{2}} + {(\frac{\sqrt{30}}{10})}^{2}}$

Simplify the expression.

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

$\sqrt{\frac{6}{25} + {(\frac{\sqrt{30}}{10})}^{2}}$

Apply the product rule to $\frac{\sqrt{30}}{10}$ .

$\sqrt{\frac{6}{25} + \frac{{\sqrt{30}}^{2}}{10^{2}}}$

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

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

$\sqrt{\frac{6}{25} + \frac{{(30^{\frac{1}{2}})}^{2}}{10^{2}}}$

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

$\sqrt{\frac{6}{25} + \frac{30^{\frac{1}{2} \cdot 2}}{10^{2}}}$

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

$\sqrt{\frac{6}{25} + \frac{30^{\frac{2}{2}}}{10^{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{\frac{6}{25} + \frac{30^{\frac{2}{2}}}{10^{2}}}$

Rewrite the expression.

$\sqrt{\frac{6}{25} + \frac{30^{1}}{10^{2}}}$

Evaluate the exponent.

$\sqrt{\frac{6}{25} + \frac{30}{10^{2}}}$

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

$\sqrt{\frac{6}{25} + \frac{30}{100}}$

Cancel the common factor of $30$ and $100$ .

Factor $10$ out of $30$ .

$\sqrt{\frac{6}{25} + \frac{10 (3)}{100}}$

Cancel the common factors.

Factor $10$ out of $100$ .

$\sqrt{\frac{6}{25} + \frac{10 \cdot 3}{10 \cdot 10}}$

Cancel the common factor.

$\sqrt{\frac{6}{25} + \frac{10 \cdot 3}{10 \cdot 10}}$

Rewrite the expression.

$\sqrt{\frac{6}{25} + \frac{3}{10}}$

To write $\frac{6}{25}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{6}{25} \cdot \frac{2}{2} + \frac{3}{10}}$

To write $\frac{3}{10}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\sqrt{\frac{6}{25} \cdot \frac{2}{2} + \frac{3}{10} \cdot \frac{5}{5}}$

Write each expression with a common denominator of $50$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{6}{25}$ by $\frac{2}{2}$ .

$\sqrt{\frac{6 \cdot 2}{25 \cdot 2} + \frac{3}{10} \cdot \frac{5}{5}}$

Multiply $25$ by $2$ .

$\sqrt{\frac{6 \cdot 2}{50} + \frac{3}{10} \cdot \frac{5}{5}}$

Multiply $\frac{3}{10}$ by $\frac{5}{5}$ .

$\sqrt{\frac{6 \cdot 2}{50} + \frac{3 \cdot 5}{10 \cdot 5}}$

Multiply $10$ by $5$ .

$\sqrt{\frac{6 \cdot 2}{50} + \frac{3 \cdot 5}{50}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{6 \cdot 2 + 3 \cdot 5}{50}}$

Simplify the numerator.

Multiply $6$ by $2$ .

$\sqrt{\frac{12 + 3 \cdot 5}{50}}$

Multiply $3$ by $5$ .

$\sqrt{\frac{12 + 15}{50}}$

Add $12$ and $15$ .

$\sqrt{\frac{27}{50}}$

Rewrite $\sqrt{\frac{27}{50}}$ as $\frac{\sqrt{27}}{\sqrt{50}}$ .

$\frac{\sqrt{27}}{\sqrt{50}}$

Simplify the numerator.

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$\frac{\sqrt{9 (3)}}{\sqrt{50}}$

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

$\frac{\sqrt{3^{2} \cdot 3}}{\sqrt{50}}$

Pull terms out from under the radical.

$\frac{3 \sqrt{3}}{\sqrt{50}}$

Simplify the denominator.

Rewrite $50$ as $5^{2} \cdot 2$ .

Factor $25$ out of $50$ .

$\frac{3 \sqrt{3}}{\sqrt{25 (2)}}$

Rewrite $25$ as $5^{2}$ .

$\frac{3 \sqrt{3}}{\sqrt{5^{2} \cdot 2}}$

Pull terms out from under the radical.

$\frac{3 \sqrt{3}}{5 \sqrt{2}}$

Multiply $\frac{3 \sqrt{3}}{5 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{3 \sqrt{3}}{5 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{3 \sqrt{3}}{5 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{3 \sqrt{3} \sqrt{2}}{5 \sqrt{2} \sqrt{2}}$

Move $\sqrt{2}$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 \sqrt{3} \sqrt{2}}{5 ({\sqrt{2}}^{1} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{3 \sqrt{3} \sqrt{2}}{5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

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

$\frac{3 \sqrt{3} \sqrt{2}}{5 {\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{3 \sqrt{3} \sqrt{2}}{5 {\sqrt{2}}^{2}}$

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

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

$\frac{3 \sqrt{3} \sqrt{2}}{5 {(2^{\frac{1}{2}})}^{2}}$

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

$\frac{3 \sqrt{3} \sqrt{2}}{5 \cdot 2^{\frac{1}{2} \cdot 2}}$

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

$\frac{3 \sqrt{3} \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 \sqrt{3} \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{3 \sqrt{3} \sqrt{2}}{5 \cdot 2^{1}}$

Evaluate the exponent.

$\frac{3 \sqrt{3} \sqrt{2}}{5 \cdot 2}$

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{3 \sqrt{2 \cdot 3}}{5 \cdot 2}$

Multiply $2$ by $3$ .

$\frac{3 \sqrt{6}}{5 \cdot 2}$

Multiply $5$ by $2$ .

$\frac{3 \sqrt{6}}{10}$

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.

$(- \frac{1}{5} + \frac{\sqrt{6}}{5}, 0)$

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.

$(- \frac{1}{5} - \frac{\sqrt{6}}{5}, 0)$

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

$(- \frac{1}{5} + \frac{\sqrt{6}}{5}, 0), (- \frac{1}{5} - \frac{\sqrt{6}}{5}, 0)$

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.

$(- \frac{1}{5} + \frac{3 \sqrt{6}}{10}, 0)$

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.

$(- \frac{1}{5} - \frac{3 \sqrt{6}}{10}, 0)$

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

$(- \frac{1}{5} + \frac{3 \sqrt{6}}{10}, 0), (- \frac{1}{5} - \frac{3 \sqrt{6}}{10}, 0)$

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.

$\frac{\sqrt{{(\frac{\sqrt{6}}{5})}^{2} + {(\frac{\sqrt{30}}{10})}^{2}}}{\frac{\sqrt{6}}{5}}$

Simplify.

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{{(\frac{\sqrt{6}}{5})}^{2} + {(\frac{\sqrt{30}}{10})}^{2}} \frac{5}{\sqrt{6}}$

Apply the product rule to $\frac{\sqrt{6}}{5}$ .

$\sqrt{\frac{{\sqrt{6}}^{2}}{5^{2}} + {(\frac{\sqrt{30}}{10})}^{2}} \frac{5}{\sqrt{6}}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sqrt{\frac{{(6^{\frac{1}{2}})}^{2}}{5^{2}} + {(\frac{\sqrt{30}}{10})}^{2}} \frac{5}{\sqrt{6}}$

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

$\sqrt{\frac{6^{\frac{1}{2} \cdot 2}}{5^{2}} + {(\frac{\sqrt{30}}{10})}^{2}} \frac{5}{\sqrt{6}}$

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

$\sqrt{\frac{6^{\frac{2}{2}}}{5^{2}} + {(\frac{\sqrt{30}}{10})}^{2}} \frac{5}{\sqrt{6}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{\frac{6^{\frac{2}{2}}}{5^{2}} + {(\frac{\sqrt{30}}{10})}^{2}} \frac{5}{\sqrt{6}}$

Rewrite the expression.

$\sqrt{\frac{6^{1}}{5^{2}} + {(\frac{\sqrt{30}}{10})}^{2}} \frac{5}{\sqrt{6}}$

Evaluate the exponent.

$\sqrt{\frac{6}{5^{2}} + {(\frac{\sqrt{30}}{10})}^{2}} \frac{5}{\sqrt{6}}$

Simplify the expression.

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

$\sqrt{\frac{6}{25} + {(\frac{\sqrt{30}}{10})}^{2}} \frac{5}{\sqrt{6}}$

Apply the product rule to $\frac{\sqrt{30}}{10}$ .

$\sqrt{\frac{6}{25} + \frac{{\sqrt{30}}^{2}}{10^{2}}} \frac{5}{\sqrt{6}}$

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

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

$\sqrt{\frac{6}{25} + \frac{{(30^{\frac{1}{2}})}^{2}}{10^{2}}} \frac{5}{\sqrt{6}}$

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

$\sqrt{\frac{6}{25} + \frac{30^{\frac{1}{2} \cdot 2}}{10^{2}}} \frac{5}{\sqrt{6}}$

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

$\sqrt{\frac{6}{25} + \frac{30^{\frac{2}{2}}}{10^{2}}} \frac{5}{\sqrt{6}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{\frac{6}{25} + \frac{30^{\frac{2}{2}}}{10^{2}}} \frac{5}{\sqrt{6}}$

Rewrite the expression.

$\sqrt{\frac{6}{25} + \frac{30^{1}}{10^{2}}} \frac{5}{\sqrt{6}}$

Evaluate the exponent.

$\sqrt{\frac{6}{25} + \frac{30}{10^{2}}} \frac{5}{\sqrt{6}}$

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

$\sqrt{\frac{6}{25} + \frac{30}{100}} \frac{5}{\sqrt{6}}$

Cancel the common factor of $30$ and $100$ .

Factor $10$ out of $30$ .

$\sqrt{\frac{6}{25} + \frac{10 (3)}{100}} \frac{5}{\sqrt{6}}$

Cancel the common factors.

Factor $10$ out of $100$ .

$\sqrt{\frac{6}{25} + \frac{10 \cdot 3}{10 \cdot 10}} \frac{5}{\sqrt{6}}$

Cancel the common factor.

$\sqrt{\frac{6}{25} + \frac{10 \cdot 3}{10 \cdot 10}} \frac{5}{\sqrt{6}}$

Rewrite the expression.

$\sqrt{\frac{6}{25} + \frac{3}{10}} \frac{5}{\sqrt{6}}$

To write $\frac{6}{25}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{6}{25} \cdot \frac{2}{2} + \frac{3}{10}} \frac{5}{\sqrt{6}}$

To write $\frac{3}{10}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\sqrt{\frac{6}{25} \cdot \frac{2}{2} + \frac{3}{10} \cdot \frac{5}{5}} \frac{5}{\sqrt{6}}$

Write each expression with a common denominator of $50$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{6}{25}$ by $\frac{2}{2}$ .

$\sqrt{\frac{6 \cdot 2}{25 \cdot 2} + \frac{3}{10} \cdot \frac{5}{5}} \frac{5}{\sqrt{6}}$

Multiply $25$ by $2$ .

$\sqrt{\frac{6 \cdot 2}{50} + \frac{3}{10} \cdot \frac{5}{5}} \frac{5}{\sqrt{6}}$

Multiply $\frac{3}{10}$ by $\frac{5}{5}$ .

$\sqrt{\frac{6 \cdot 2}{50} + \frac{3 \cdot 5}{10 \cdot 5}} \frac{5}{\sqrt{6}}$

Multiply $10$ by $5$ .

$\sqrt{\frac{6 \cdot 2}{50} + \frac{3 \cdot 5}{50}} \frac{5}{\sqrt{6}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{6 \cdot 2 + 3 \cdot 5}{50}} \frac{5}{\sqrt{6}}$

Simplify the numerator.

Multiply $6$ by $2$ .

$\sqrt{\frac{12 + 3 \cdot 5}{50}} \frac{5}{\sqrt{6}}$

Multiply $3$ by $5$ .

$\sqrt{\frac{12 + 15}{50}} \frac{5}{\sqrt{6}}$

Add $12$ and $15$ .

$\sqrt{\frac{27}{50}} \frac{5}{\sqrt{6}}$

Rewrite $\sqrt{\frac{27}{50}}$ as $\frac{\sqrt{27}}{\sqrt{50}}$ .

$\frac{\sqrt{27}}{\sqrt{50}} \cdot \frac{5}{\sqrt{6}}$

Simplify the numerator.

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$\frac{\sqrt{9 (3)}}{\sqrt{50}} \cdot \frac{5}{\sqrt{6}}$

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

$\frac{\sqrt{3^{2} \cdot 3}}{\sqrt{50}} \cdot \frac{5}{\sqrt{6}}$

Pull terms out from under the radical.

$\frac{3 \sqrt{3}}{\sqrt{50}} \cdot \frac{5}{\sqrt{6}}$

Simplify the denominator.

Rewrite $50$ as $5^{2} \cdot 2$ .

Factor $25$ out of $50$ .

$\frac{3 \sqrt{3}}{\sqrt{25 (2)}} \cdot \frac{5}{\sqrt{6}}$

Rewrite $25$ as $5^{2}$ .

$\frac{3 \sqrt{3}}{\sqrt{5^{2} \cdot 2}} \cdot \frac{5}{\sqrt{6}}$

Pull terms out from under the radical.

$\frac{3 \sqrt{3}}{5 \sqrt{2}} \cdot \frac{5}{\sqrt{6}}$

Simplify terms.

Cancel the common factor of $5$ .

Factor $5$ out of $5 \sqrt{2}$ .

$\frac{3 \sqrt{3}}{5 (\sqrt{2})} \cdot \frac{5}{\sqrt{6}}$

Cancel the common factor.

$\frac{3 \sqrt{3}}{5 \sqrt{2}} \cdot \frac{5}{\sqrt{6}}$

Rewrite the expression.

$\frac{3 \sqrt{3}}{\sqrt{2}} \cdot \frac{1}{\sqrt{6}}$

Multiply $\frac{3 \sqrt{3}}{\sqrt{2}}$ by $\frac{1}{\sqrt{6}}$ .

$\frac{3 \sqrt{3}}{\sqrt{2} \sqrt{6}}$

Combine using the product rule for radicals.

$\frac{3 \sqrt{3}}{\sqrt{2 \cdot 6}}$

Multiply $2$ by $6$ .

$\frac{3 \sqrt{3}}{\sqrt{12}}$

Combine $\sqrt{3}$ and $\sqrt{12}$ into a single radical.

$3 \sqrt{\frac{3}{12}}$

Cancel the common factor of $3$ and $12$ .

Factor $3$ out of $3$ .

$3 \sqrt{\frac{3 (1)}{12}}$

Cancel the common factors.

Factor $3$ out of $12$ .

$3 \sqrt{\frac{3 \cdot 1}{3 \cdot 4}}$

Cancel the common factor.

$3 \sqrt{\frac{3 \cdot 1}{3 \cdot 4}}$

Rewrite the expression.

$3 \sqrt{\frac{1}{4}}$

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$3 \frac{\sqrt{1}}{\sqrt{4}}$

Any root of $1$ is $1$ .

$3 \frac{1}{\sqrt{4}}$

Simplify the denominator.

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

$3 \frac{1}{\sqrt{2^{2}}}$

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

$3 (\frac{1}{2})$

Combine $3$ and $\frac{1}{2}$ .

$\frac{3}{2}$

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.

$\frac{\frac{\frac{\sqrt{30}}{10^{2}}}{3 \sqrt{6}}}{10}$

Simplify.

Multiply the numerator by the reciprocal of the denominator.

$\frac{\frac{\sqrt{30}}{10^{2}}}{3 \sqrt{6}} \cdot \frac{1}{10}$

Combine.

$\frac{\frac{\sqrt{30}}{10^{2}} \cdot 1}{3 \sqrt{6} \cdot 10}$

Simplify the expression.

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

$\frac{\frac{\sqrt{30}}{100} \cdot 1}{3 \sqrt{6} \cdot 10}$

Multiply $\frac{\sqrt{30}}{100}$ by $1$ .

$\frac{\frac{\sqrt{30}}{100}}{3 \sqrt{6} \cdot 10}$

Multiply $10$ by $3$ .

$\frac{\frac{\sqrt{30}}{100}}{30 \sqrt{6}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{30}}{100} \cdot \frac{1}{30 \sqrt{6}}$

Multiply $\frac{1}{30 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{30}}{100} (\frac{1}{30 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

Combine and simplify the denominator.

Multiply $\frac{1}{30 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{30}}{100} \cdot \frac{\sqrt{6}}{30 \sqrt{6} \sqrt{6}}$

Move $\sqrt{6}$ .

$\frac{\sqrt{30}}{100} \cdot \frac{\sqrt{6}}{30 (\sqrt{6} \sqrt{6})}$

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{30}}{100} \cdot \frac{\sqrt{6}}{30 ({\sqrt{6}}^{1} \sqrt{6})}$

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{30}}{100} \cdot \frac{\sqrt{6}}{30 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}$

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

$\frac{\sqrt{30}}{100} \cdot \frac{\sqrt{6}}{30 {\sqrt{6}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{\sqrt{30}}{100} \cdot \frac{\sqrt{6}}{30 {\sqrt{6}}^{2}}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{\sqrt{30}}{100} \cdot \frac{\sqrt{6}}{30 {(6^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt{30}}{100} \cdot \frac{\sqrt{6}}{30 \cdot 6^{\frac{1}{2} \cdot 2}}$

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

$\frac{\sqrt{30}}{100} \cdot \frac{\sqrt{6}}{30 \cdot 6^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{30}}{100} \cdot \frac{\sqrt{6}}{30 \cdot 6^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{\sqrt{30}}{100} \cdot \frac{\sqrt{6}}{30 \cdot 6^{1}}$

Evaluate the exponent.

$\frac{\sqrt{30}}{100} \cdot \frac{\sqrt{6}}{30 \cdot 6}$

Multiply $30$ by $6$ .

$\frac{\sqrt{30}}{100} \cdot \frac{\sqrt{6}}{180}$

Multiply $\frac{\sqrt{30}}{100} \cdot \frac{\sqrt{6}}{180}$ .

Multiply $\frac{\sqrt{30}}{100}$ by $\frac{\sqrt{6}}{180}$ .

$\frac{\sqrt{30} \sqrt{6}}{100 \cdot 180}$

Combine using the product rule for radicals.

$\frac{\sqrt{30 \cdot 6}}{100 \cdot 180}$

Multiply $30$ by $6$ .

$\frac{\sqrt{180}}{100 \cdot 180}$

Multiply $100$ by $180$ .

$\frac{\sqrt{180}}{18000}$

Simplify the numerator.

Rewrite $180$ as $6^{2} \cdot 5$ .

Factor $36$ out of $180$ .

$\frac{\sqrt{36 (5)}}{18000}$

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

$\frac{\sqrt{6^{2} \cdot 5}}{18000}$

Pull terms out from under the radical.

$\frac{6 \sqrt{5}}{18000}$

Cancel the common factor of $6$ and $18000$ .

Factor $6$ out of $6 \sqrt{5}$ .

$\frac{6 (\sqrt{5})}{18000}$

Cancel the common factors.

Factor $6$ out of $18000$ .

$\frac{6 \sqrt{5}}{6 \cdot 3000}$

Cancel the common factor.

$\frac{6 \sqrt{5}}{6 \cdot 3000}$

Rewrite the expression.

$\frac{\sqrt{5}}{3000}$

Step 10

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

$y = \pm \frac{\sqrt{5}}{2} \cdot (x - (- \frac{1}{5})) + 0$

Step 11

Simplify to find the first asymptote.

Multiply $- 1$ by $- \frac{1}{5}$ .

$y = \frac{\sqrt{5}}{2} \cdot (x + \frac{1}{5}) + 0$

Remove parentheses.

$y = \frac{\sqrt{5}}{2} \cdot (x - (- \frac{1}{5})) + 0$

Simplify $\frac{\sqrt{5}}{2} \cdot (x - (- \frac{1}{5})) + 0$ .

Add $\frac{\sqrt{5}}{2} \cdot (x - (- \frac{1}{5}))$ and $0$ .

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

Multiply $- (- \frac{1}{5})$ .

Multiply $- 1$ by $- 1$ .

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

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

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

Apply the distributive property.

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

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

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

Multiply $\frac{\sqrt{5}}{2} \cdot \frac{1}{5}$ .

Multiply $\frac{\sqrt{5}}{2}$ by $\frac{1}{5}$ .

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

Multiply $2$ by $5$ .

$y = \frac{\sqrt{5} x}{2} + \frac{\sqrt{5}}{10}$

Step 12

Simplify to find the second asymptote.

Multiply $- 1$ by $- \frac{1}{5}$ .

$y = - \frac{\sqrt{5}}{2} \cdot (x + \frac{1}{5}) + 0$

Remove parentheses.

$y = - \frac{\sqrt{5}}{2} \cdot (x - (- \frac{1}{5})) + 0$

Simplify $- \frac{\sqrt{5}}{2} \cdot (x - (- \frac{1}{5})) + 0$ .

Add $- \frac{\sqrt{5}}{2} \cdot (x - (- \frac{1}{5}))$ and $0$ .

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

Multiply $- (- \frac{1}{5})$ .

Multiply $- 1$ by $- 1$ .

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

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

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

Apply the distributive property.

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

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

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

Multiply $- \frac{\sqrt{5}}{2} \cdot \frac{1}{5}$ .

Multiply $\frac{1}{5}$ by $\frac{\sqrt{5}}{2}$ .

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

Multiply $5$ by $2$ .

$y = - \frac{x \sqrt{5}}{2} - \frac{\sqrt{5}}{10}$

Step 13

This hyperbola has two asymptotes.

$y = \frac{\sqrt{5} x}{2} + \frac{\sqrt{5}}{10}, y = - \frac{x \sqrt{5}}{2} - \frac{\sqrt{5}}{10}$

Step 14

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

Center: $(- \frac{1}{5}, 0)$

Vertices: $(- \frac{1}{5} + \frac{\sqrt{6}}{5}, 0), (- \frac{1}{5} - \frac{\sqrt{6}}{5}, 0)$

Foci: $(- \frac{1}{5} + \frac{3 \sqrt{6}}{10}, 0), (- \frac{1}{5} - \frac{3 \sqrt{6}}{10}, 0)$

Eccentricity: $\frac{3}{2}$

Focal Parameter: $\frac{\sqrt{5}}{3000}$

Asymptotes: $y = \frac{\sqrt{5} x}{2} + \frac{\sqrt{5}}{10}$ , $y = - \frac{x \sqrt{5}}{2} - \frac{\sqrt{5}}{10}$

Step 15

Find the Axis of Symmetry 5x^2-4y^2+2x-1=0

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