Find the Foci 64x^2+81y^2=5184

$64 x^{2} + 81 y^{2} = 5184$

Step 1

Find the standard form of the ellipse.

Divide each term by $5184$ to make the right side equal to one.

$\frac{64 x^{2}}{5184} + \frac{81 y^{2}}{5184} = \frac{5184}{5184}$

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 $1$ .

$\frac{x^{2}}{81} + \frac{y^{2}}{64} = 1$

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.

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

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.

$a = 9$

$b = 8$

$k = 0$

$h = 0$

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.

$\sqrt{{(9)}^{2} - {(8)}^{2}}$

Simplify.

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

$\sqrt{81 - {(8)}^{2}}$

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

$\sqrt{81 - 1 \cdot 64}$

Multiply $- 1$ by $64$ .

$\sqrt{81 - 64}$

Subtract $64$ from $81$ .

$\sqrt{17}$

Step 5

Find the foci.

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

$(h + c, k)$

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

$(0 + \sqrt{17}, 0)$

Simplify.

$(\sqrt{17}, 0)$

The second focus of an ellipse can be found by subtracting $c$ from $h$ .

$(h - c, k)$

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

$(0 - (\sqrt{17}), 0)$

Simplify.

$(- \sqrt{17}, 0)$

Ellipses have two foci.

${Focus}_{1}$ : $(\sqrt{17}, 0)$

${Focus}_{2}$ : $(- \sqrt{17}, 0)$

${Focus}_{1}$ : $(\sqrt{17}, 0)$

${Focus}_{2}$ : $(- \sqrt{17}, 0)$

Step 6

Find the Foci 64x^2+81y^2=5184

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