Find the Foci ((y-2)^2)/64-((x-4)^2)/36=1

$\frac{{(y - 2)}^{2}}{64} - \frac{{(x - 4)}^{2}}{36} = 1$

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 $1$ .

$\frac{{(y - 2)}^{2}}{64} - \frac{{(x - 4)}^{2}}{36} = 1$

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

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

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, $a$ .

$a = 8$

$b = 6$

$k = 2$

$h = 4$

Find $c$ , the distance from the center to a focus.

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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.

$\sqrt{{(8)}^{2} + {(6)}^{2}}$

Simplify.

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Raise $8$ to the power of $2$ .

$\sqrt{64 + {(6)}^{2}}$

Raise $6$ to the power of $2$ .

$\sqrt{64 + 36}$

Add $64$ and $36$ .

$\sqrt{100}$

Rewrite $100$ as $10^{2}$ .

$\sqrt{10^{2}}$

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

$10$

Find the foci.

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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.

$(4, 12)$

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.

$(4, - 8)$

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

$(4, 12), (4, - 8)$

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