Find the Foci 9y^2-x^2=81

$9 y^{2} - x^{2} = 81$

Step 1

Find the standard form of the hyperbola.

Divide each term by $81$ to make the right side equal to one.

$\frac{9 y^{2}}{81} - \frac{x^{2}}{81} = \frac{81}{81}$

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 $1$ .

$\frac{y^{2}}{9} - \frac{x^{2}}{81} = 1$

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, $a$ .

$a = 3$

$b = 9$

$k = 0$

$h = 0$

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.

$\sqrt{{(3)}^{2} + {(9)}^{2}}$

Simplify.

Raise $3$ to the power of $2$ .

$\sqrt{9 + {(9)}^{2}}$

Raise $9$ to the power of $2$ .

$\sqrt{9 + 81}$

Add $9$ and $81$ .

$\sqrt{90}$

Rewrite $90$ as $3^{2} \cdot 10$ .

Factor $9$ out of $90$ .

$\sqrt{9 (10)}$

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2} \cdot 10}$

Pull terms out from under the radical.

$3 \sqrt{10}$

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.

$(0, 3 \sqrt{10})$

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.

$(0, - 3 \sqrt{10})$

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

$(0, 3 \sqrt{10}), (0, - 3 \sqrt{10})$

Step 6

Find the Foci 9y^2-x^2=81

Related Questions

Support LCMGCF.com by disabling your adblocker.