Find the Foci y^2-25x^2=25

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

Step 1

Find the standard form of the hyperbola.

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

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

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 $1$ .

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

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.

$\frac{{(y - k)}^{2}}{a^{2}} - \frac{{(x - h)}^{2}}{b^{2}} = 1$

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 $c$ , 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.

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

Simplify.

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

$\sqrt{25 + {(1)}^{2}}$

One to any power is one.

$\sqrt{25 + 1}$

Add $25$ and $1$ .

$\sqrt{26}$

Step 5

Find the foci.

The first focus of a hyperbola can be found by adding $c$ to $k$ .

$(h, k + c)$

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

$(0, \sqrt{26})$

The second focus of a hyperbola can be found by subtracting $c$ from $k$ .

$(h, k - c)$

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

$(0, - \sqrt{26})$

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

$(0, \sqrt{26}), (0, - \sqrt{26})$

Step 6

Find the Foci y^2-25x^2=25

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