Precalculus Examples

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

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 $1$ .

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

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.

$a = 9$

$b = 8$

$k = 0$

$h = 0$

Step 4

Find the eccentricity by using the following formula.

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

Step 5

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

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

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

$\frac{\sqrt{81 - 8^{2}}}{9}$

Step 6.2

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

$\frac{\sqrt{81 - 1 \cdot 64}}{9}$

Step 6.3

Multiply $- 1$ by $64$ .

$\frac{\sqrt{81 - 64}}{9}$

Step 6.4

Subtract $64$ from $81$ .

$\frac{\sqrt{17}}{9}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{17}}{9}$

Decimal Form:

$0.45812284 \dots$

Step 8