Graph ((x-2)^2)/25-((y-3)^2)/36=1

$\frac{{(x - 2)}^{2}}{25} - \frac{{(y - 3)}^{2}}{36} = 1$

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 $1$ .

$\frac{{(x - 2)}^{2}}{25} - \frac{{(y - 3)}^{2}}{36} = 1$

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, $a$ .

$a = 5$

$b = 6$

$k = 3$

$h = 2$

Step 4

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

$(2, 3)$

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.

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

Simplify.

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

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

Raise $6$ to the power of $2$ .

$\sqrt{25 + 36}$

Add $25$ and $36$ .

$\sqrt{61}$

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.

$(7, 3)$

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.

$(- 3, 3)$

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

$(7, 3), (- 3, 3)$

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.

$(2 + \sqrt{61}, 3)$

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.

$(2 - \sqrt{61}, 3)$

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

$(2 + \sqrt{61}, 3), (2 - \sqrt{61}, 3)$

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.

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

Simplify the numerator.

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

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

Raise $6$ to the power of $2$ .

$\frac{\sqrt{25 + 36}}{5}$

Add $25$ and $36$ .

$\frac{\sqrt{61}}{5}$

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.

$\frac{6^{2}}{\sqrt{61}}$

Simplify.

Raise $6$ to the power of $2$ .

$\frac{36}{\sqrt{61}}$

Multiply $\frac{36}{\sqrt{61}}$ by $\frac{\sqrt{61}}{\sqrt{61}}$ .

$\frac{36}{\sqrt{61}} \cdot \frac{\sqrt{61}}{\sqrt{61}}$

Combine and simplify the denominator.

Multiply $\frac{36}{\sqrt{61}}$ by $\frac{\sqrt{61}}{\sqrt{61}}$ .

$\frac{36 \sqrt{61}}{\sqrt{61} \sqrt{61}}$

Raise $\sqrt{61}$ to the power of $1$ .

$\frac{36 \sqrt{61}}{{\sqrt{61}}^{1} \sqrt{61}}$

Raise $\sqrt{61}$ to the power of $1$ .

$\frac{36 \sqrt{61}}{{\sqrt{61}}^{1} {\sqrt{61}}^{1}}$

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

$\frac{36 \sqrt{61}}{{\sqrt{61}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{36 \sqrt{61}}{{\sqrt{61}}^{2}}$

Rewrite ${\sqrt{61}}^{2}$ as $61$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{61}$ as $61^{\frac{1}{2}}$ .

$\frac{36 \sqrt{61}}{{(61^{\frac{1}{2}})}^{2}}$

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

$\frac{36 \sqrt{61}}{61^{\frac{1}{2} \cdot 2}}$

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

$\frac{36 \sqrt{61}}{61^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{36 \sqrt{61}}{61^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{36 \sqrt{61}}{61^{1}}$

Evaluate the exponent.

$\frac{36 \sqrt{61}}{61}$

Step 10

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

$y = \pm \frac{6}{5} \cdot (x - (2)) + 3$

Step 11

Simplify to find the first asymptote.

Remove parentheses.

$y = \frac{6}{5} \cdot (x - (2)) + 3$

Simplify $\frac{6}{5} \cdot (x - (2)) + 3$ .

Simplify each term.

Multiply $- 1$ by $2$ .

$y = \frac{6}{5} \cdot (x - 2) + 3$

Apply the distributive property.

$y = \frac{6}{5} x + \frac{6}{5} \cdot - 2 + 3$

Combine $\frac{6}{5}$ and $x$ .

$y = \frac{6 x}{5} + \frac{6}{5} \cdot - 2 + 3$

Multiply $\frac{6}{5} \cdot - 2$ .

Combine $\frac{6}{5}$ and $- 2$ .

$y = \frac{6 x}{5} + \frac{6 \cdot - 2}{5} + 3$

Multiply $6$ by $- 2$ .

$y = \frac{6 x}{5} + \frac{- 12}{5} + 3$

Move the negative in front of the fraction.

$y = \frac{6 x}{5} - \frac{12}{5} + 3$

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

$y = \frac{6 x}{5} - \frac{12}{5} + 3 \cdot \frac{5}{5}$

Combine $3$ and $\frac{5}{5}$ .

$y = \frac{6 x}{5} - \frac{12}{5} + \frac{3 \cdot 5}{5}$

Combine the numerators over the common denominator.

$y = \frac{6 x}{5} + \frac{- 12 + 3 \cdot 5}{5}$

Simplify the numerator.

Multiply $3$ by $5$ .

$y = \frac{6 x}{5} + \frac{- 12 + 15}{5}$

Add $- 12$ and $15$ .

$y = \frac{6 x}{5} + \frac{3}{5}$

Step 12

Simplify to find the second asymptote.

Remove parentheses.

$y = - \frac{6}{5} \cdot (x - (2)) + 3$

Simplify $- \frac{6}{5} \cdot (x - (2)) + 3$ .

Simplify each term.

Multiply $- 1$ by $2$ .

$y = - \frac{6}{5} \cdot (x - 2) + 3$

Apply the distributive property.

$y = - \frac{6}{5} x - \frac{6}{5} \cdot - 2 + 3$

Combine $x$ and $\frac{6}{5}$ .

$y = - \frac{x \cdot 6}{5} - \frac{6}{5} \cdot - 2 + 3$

Multiply $- \frac{6}{5} \cdot - 2$ .

Multiply $- 2$ by $- 1$ .

$y = - \frac{x \cdot 6}{5} + 2 (\frac{6}{5}) + 3$

Combine $2$ and $\frac{6}{5}$ .

$y = - \frac{x \cdot 6}{5} + \frac{2 \cdot 6}{5} + 3$

Multiply $2$ by $6$ .

$y = - \frac{x \cdot 6}{5} + \frac{12}{5} + 3$

Move $6$ to the left of $x$ .

$y = - \frac{6 x}{5} + \frac{12}{5} + 3$

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

$y = - \frac{6 x}{5} + \frac{12}{5} + 3 \cdot \frac{5}{5}$

Combine $3$ and $\frac{5}{5}$ .

$y = - \frac{6 x}{5} + \frac{12}{5} + \frac{3 \cdot 5}{5}$

Combine the numerators over the common denominator.

$y = - \frac{6 x}{5} + \frac{12 + 3 \cdot 5}{5}$

Simplify the numerator.

Multiply $3$ by $5$ .

$y = - \frac{6 x}{5} + \frac{12 + 15}{5}$

Add $12$ and $15$ .

$y = - \frac{6 x}{5} + \frac{27}{5}$

Step 13

This hyperbola has two asymptotes.

$y = \frac{6 x}{5} + \frac{3}{5}, y = - \frac{6 x}{5} + \frac{27}{5}$

Step 14

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

Center: $(2, 3)$

Vertices: $(7, 3), (- 3, 3)$

Foci: $(2 + \sqrt{61}, 3), (2 - \sqrt{61}, 3)$

Eccentricity: $\frac{\sqrt{61}}{5}$

Focal Parameter: $\frac{36 \sqrt{61}}{61}$

Asymptotes: $y = \frac{6 x}{5} + \frac{3}{5}$ , $y = - \frac{6 x}{5} + \frac{27}{5}$

Step 15

Graph ((x-2)^2)/25-((y-3)^2)/36=1

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