Find the Foci (x^2)/(16/9)+(y^2)/(25/9)=1

$\frac{x^{2}}{\frac{16}{9}} + \frac{y^{2}}{\frac{25}{9}} = 1$

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 $1$ .

$\frac{x^{2}}{\frac{16}{9}} + \frac{y^{2}}{\frac{25}{9}} = 1$

Step 2

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

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

Step 3

Match the values in this ellipse to those of the standard form. The variable $a$ represents the radius of the major axis of the ellipse, $b$ represents the radius of the minor axis of the ellipse, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from the origin.

$a = \frac{5}{3}$

$b = \frac{4}{3}$

$k = 0$

$h = 0$

Step 4

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

Find the distance from the center to a focus of the ellipse by using the following formula.

$\sqrt{a^{2} - b^{2}}$

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(\frac{5}{3})}^{2} - {(\frac{4}{3})}^{2}}$

Simplify.

Apply the product rule to $\frac{5}{3}$ .

$\sqrt{\frac{5^{2}}{3^{2}} - {(\frac{4}{3})}^{2}}$

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

$\sqrt{\frac{25}{3^{2}} - {(\frac{4}{3})}^{2}}$

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

$\sqrt{\frac{25}{9} - {(\frac{4}{3})}^{2}}$

Apply the product rule to $\frac{4}{3}$ .

$\sqrt{\frac{25}{9} - \frac{4^{2}}{3^{2}}}$

Raise $4$ to the power of $2$ .

$\sqrt{\frac{25}{9} - \frac{16}{3^{2}}}$

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

$\sqrt{\frac{25}{9} - \frac{16}{9}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{25 - 16}{9}}$

Subtract $16$ from $25$ .

$\sqrt{\frac{9}{9}}$

Divide $9$ by $9$ .

$\sqrt{1}$

Any root of $1$ is $1$ .

$1$

Step 5

Find the foci.

The first focus of an ellipse can be found by adding $c$ to $k$ .

$(h, k + c)$

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(0, 0 + 1)$

Simplify.

$(0, 1)$

The first focus of an ellipse can be found by subtracting $c$ from $k$ .

$(h, k - c)$

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(0, 0 - (1))$

Simplify.

$(0, - 1)$

Ellipses have two foci.

${Focus}_{1}$ : $(0, 1)$

${Focus}_{2}$ : $(0, - 1)$

${Focus}_{1}$ : $(0, 1)$

${Focus}_{2}$ : $(0, - 1)$

Step 6

Find the Foci (x^2)/(16/9)+(y^2)/(25/9)=1

Related Questions

Support LCMGCF.com by disabling your adblocker.