Trigonometry Examples

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

Step 1

Find the standard form of the hyperbola.

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Step 1.1

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

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

Step 1.2

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 $1$ .

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

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, $a$ .

$a = 5$

$b = 1$

$k = 0$

$h = 0$

Step 4

Find the vertices.

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Step 4.1

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

$(h + a, k)$

Step 4.2

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

$(5, 0)$

Step 4.3

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

$(h - a, k)$

Step 4.4

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

$(- 5, 0)$

Step 4.5

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

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

Step 5