Find the Focus x^2=4py

$x^{2} = 4 p y$

Find the standard form of the hyperbola.

Move all terms containing variables to the left side of the equation.

Subtract $4 p y$ from both sides of the equation.

$x^{2} - 4 p y = 0$

Reorder $x^{2}$ and $- 4 p y$ .

$- 4 p y + x^{2} = 0$

Divide each term by $0$ to make the right side equal to one.

$- \frac{4 p y}{0} + \frac{x^{2}}{0} = \frac{0}{0}$

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 $1$ .

$x^{2} - p y = 1$

This is the form of a hyperbola. Use this form to determine the values used to find vertices and asymptotes of the hyperbola.

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

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, $a$ .

$a = 1$

$b = 1$

$k = 0$

$h = 0$

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.

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

Simplify.

One to any power is one.

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

One to any power is one.

$\sqrt{1 + 1}$

Add $1$ and $1$ .

$\sqrt{2}$

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.

$(\sqrt{2}, 0)$

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.

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

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

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

Find the Focus x^2=4py

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