Find the Foci ((y+2)^2)/25-((x-3)^2)/4=1

$\frac{{(y + 2)}^{2}}{25} - \frac{{(x - 3)}^{2}}{4} = 1$

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 $1$ .

$\frac{{(y + 2)}^{2}}{25} - \frac{{(x - 3)}^{2}}{4} = 1$

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, $a$ .

$a = 5$

$b = 2$

$k = - 2$

$h = 3$

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.

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

Simplify.

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

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

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

$\sqrt{25 + 4}$

Add $25$ and $4$ .

$\sqrt{29}$

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.

$(3, - 2 + \sqrt{29})$

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.

$(3, - 2 - \sqrt{29})$

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

$(3, - 2 + \sqrt{29}), (3, - 2 - \sqrt{29})$

Step 6

Find the Foci ((y+2)^2)/25-((x-3)^2)/4=1

Related Questions

Support LCMGCF.com by disabling your adblocker.