Find the Eccentricity (x^2)/9+(y^2)/81=1

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

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 $1$ .

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

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.

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

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.

$a = 9$

$b = 3$

$k = 0$

$h = 0$

Find the eccentricity by using the following formula.

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

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

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

Simplify.

Simplify the numerator.

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

$\frac{\sqrt{81 - 3^{2}}}{9}$

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

$\frac{\sqrt{81 - 1 \cdot 9}}{9}$

Multiply $- 1$ by $9$ .

$\frac{\sqrt{81 - 9}}{9}$

Subtract $9$ from $81$ .

$\frac{\sqrt{72}}{9}$

Rewrite $72$ as $6^{2} \cdot 2$ .

Factor $36$ out of $72$ .

$\frac{\sqrt{36 (2)}}{9}$

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{6^{2} \cdot 2}}{9}$

Pull terms out from under the radical.

$\frac{6 \sqrt{2}}{9}$

Cancel the common factor of $6$ and $9$ .

Factor $3$ out of $6 \sqrt{2}$ .

$\frac{3 (2 \sqrt{2})}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$\frac{3 (2 \sqrt{2})}{3 (3)}$

Cancel the common factor.

$\frac{3 (2 \sqrt{2})}{3 \cdot 3}$

Rewrite the expression.

$\frac{2 \sqrt{2}}{3}$

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{2}}{3}$

Decimal Form:

$0.94280904 \dots$

Find the Eccentricity (x^2)/9+(y^2)/81=1

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