Graph 25x^2-16y^2=400

$25 x^{2} - 16 y^{2} = 400$

Step 1

Find the standard form of the hyperbola.

Divide each term by $400$ to make the right side equal to one.

$\frac{25 x^{2}}{400} - \frac{16 y^{2}}{400} = \frac{400}{400}$

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 $1$ .

$\frac{x^{2}}{16} - \frac{y^{2}}{25} = 1$

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, $a$ .

$a = 4$

$b = 5$

$k = 0$

$h = 0$

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.

$\sqrt{{(4)}^{2} + {(5)}^{2}}$

Simplify.

Raise $4$ to the power of $2$ .

$\sqrt{16 + {(5)}^{2}}$

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

$\sqrt{16 + 25}$

Add $16$ and $25$ .

$\sqrt{41}$

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.

$(4, 0)$

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.

$(- 4, 0)$

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

$(4, 0), (- 4, 0)$

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.

$(\sqrt{41}, 0)$

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.

$(- \sqrt{41}, 0)$

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

$(\sqrt{41}, 0), (- \sqrt{41}, 0)$

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.

$\frac{\sqrt{{(4)}^{2} + {(5)}^{2}}}{4}$

Simplify the numerator.

Raise $4$ to the power of $2$ .

$\frac{\sqrt{16 + 5^{2}}}{4}$

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

$\frac{\sqrt{16 + 25}}{4}$

Add $16$ and $25$ .

$\frac{\sqrt{41}}{4}$

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.

$\frac{5^{2}}{\sqrt{41}}$

Simplify.

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

$\frac{25}{\sqrt{41}}$

Multiply $\frac{25}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\frac{25}{\sqrt{41}} \cdot \frac{\sqrt{41}}{\sqrt{41}}$

Combine and simplify the denominator.

Multiply $\frac{25}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\frac{25 \sqrt{41}}{\sqrt{41} \sqrt{41}}$

Raise $\sqrt{41}$ to the power of $1$ .

$\frac{25 \sqrt{41}}{{\sqrt{41}}^{1} \sqrt{41}}$

Raise $\sqrt{41}$ to the power of $1$ .

$\frac{25 \sqrt{41}}{{\sqrt{41}}^{1} {\sqrt{41}}^{1}}$

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

$\frac{25 \sqrt{41}}{{\sqrt{41}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{25 \sqrt{41}}{{\sqrt{41}}^{2}}$

Rewrite ${\sqrt{41}}^{2}$ as $41$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{41}$ as $41^{\frac{1}{2}}$ .

$\frac{25 \sqrt{41}}{{(41^{\frac{1}{2}})}^{2}}$

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

$\frac{25 \sqrt{41}}{41^{\frac{1}{2} \cdot 2}}$

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

$\frac{25 \sqrt{41}}{41^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{25 \sqrt{41}}{41^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{25 \sqrt{41}}{41^{1}}$

Evaluate the exponent.

$\frac{25 \sqrt{41}}{41}$

Step 10

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

$y = \pm \frac{5}{4} x + 0$

Step 11

Simplify $\frac{5}{4} x + 0$ .

Add $\frac{5}{4} x$ and $0$ .

$y = \frac{5}{4} x$

Combine $\frac{5}{4}$ and $x$ .

$y = \frac{5 x}{4}$

Step 12

Simplify $- \frac{5}{4} x + 0$ .

Add $- \frac{5}{4} x$ and $0$ .

$y = - \frac{5}{4} x$

Combine $x$ and $\frac{5}{4}$ .

$y = - \frac{x \cdot 5}{4}$

Move $5$ to the left of $x$ .

$y = - \frac{5 x}{4}$

Step 13

This hyperbola has two asymptotes.

$y = \frac{5 x}{4}, y = - \frac{5 x}{4}$

Step 14

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

Center: $(0, 0)$

Vertices: $(4, 0), (- 4, 0)$

Foci: $(\sqrt{41}, 0), (- \sqrt{41}, 0)$

Eccentricity: $\frac{\sqrt{41}}{4}$

Focal Parameter: $\frac{25 \sqrt{41}}{41}$

Asymptotes: $y = \frac{5 x}{4}$ , $y = - \frac{5 x}{4}$

Step 15

Graph 25x^2-16y^2=400

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