Find the Foci (y^2)/225-(x^2)/400=1

$\frac{y^{2}}{225} - \frac{x^{2}}{400} = 1$

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 $1$ .

$\frac{y^{2}}{225} - \frac{x^{2}}{400} = 1$

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, $a$ .

$a = 15$

$b = 20$

$k = 0$

$h = 0$

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.

$\sqrt{{(15)}^{2} + {(20)}^{2}}$

Simplify.

Raise $15$ to the power of $2$ .

$\sqrt{225 + {(20)}^{2}}$

Raise $20$ to the power of $2$ .

$\sqrt{225 + 400}$

Add $225$ and $400$ .

$\sqrt{625}$

Rewrite $625$ as $25^{2}$ .

$\sqrt{25^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$25$

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.

$(0, 25)$

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.

$(0, - 25)$

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

$(0, 25), (0, - 25)$

Step 6

Find the Foci (y^2)/225-(x^2)/400=1

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