Graph x^2-y^2=25

$x^{2} - y^{2} = 25$

Step 1

Find the standard form of the hyperbola.

Divide each term by $25$ to make the right side equal to one.

$\frac{x^{2}}{25} - \frac{y^{2}}{25} = \frac{25}{25}$

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 $1$ .

$\frac{x^{2}}{25} - \frac{y^{2}}{25} = 1$

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, $a$ .

$a = 5$

$b = 5$

$k = 0$

$h = 0$

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.

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

Simplify.

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

$\sqrt{25 + {(5)}^{2}}$

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

$\sqrt{25 + 25}$

Add $25$ and $25$ .

$\sqrt{50}$

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

Factor $25$ out of $50$ .

$\sqrt{25 (2)}$

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

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

Pull terms out from under the radical.

$5 \sqrt{2}$

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.

$(5, 0)$

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.

$(- 5, 0)$

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

$(5, 0), (- 5, 0)$

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.

$(5 \sqrt{2}, 0)$

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.

$(- 5 \sqrt{2}, 0)$

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

$(5 \sqrt{2}, 0), (- 5 \sqrt{2}, 0)$

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.

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

Simplify.

Simplify the numerator.

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

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

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

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

Add $25$ and $25$ .

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

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

Factor $25$ out of $50$ .

$\frac{\sqrt{25 (2)}}{5}$

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

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

Pull terms out from under the radical.

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $\sqrt{2}$ by $1$ .

$\sqrt{2}$

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.

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

Simplify.

Cancel the common factor of $5^{2}$ and $5$ .

Factor $5$ out of $5^{2}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 10

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

$y = \pm 1 \cdot x + 0$

Step 11

Simplify $1 \cdot x + 0$ .

Add $1 \cdot x$ and $0$ .

$y = 1 \cdot x$

Multiply $x$ by $1$ .

$y = x$

Step 12

Simplify $- 1 \cdot x + 0$ .

Add $- 1 \cdot x$ and $0$ .

$y = - 1 \cdot x$

Rewrite $- 1 x$ as $- x$ .

$y = - x$

Step 13

This hyperbola has two asymptotes.

$y = x, y = - x$

Step 14

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

Center: $(0, 0)$

Vertices: $(5, 0), (- 5, 0)$

Foci: $(5 \sqrt{2}, 0), (- 5 \sqrt{2}, 0)$

Eccentricity: $\sqrt{2}$

Focal Parameter: $\frac{5 \sqrt{2}}{2}$

Asymptotes: $y = x$ , $y = - x$

Step 15

Graph x^2-y^2=25

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