Find the Eccentricity ((x+1)^2)/9+((y-5)^2)/64=1

$\frac{{(x + 1)}^{2}}{9} + \frac{{(y - 5)}^{2}}{64} = 1$

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 $1$ .

$\frac{{(x + 1)}^{2}}{9} + \frac{{(y - 5)}^{2}}{64} = 1$

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.

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

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.

$a = 8$

$b = 3$

$k = 5$

$h = - 1$

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.

$\frac{\sqrt{{(8)}^{2} - {(3)}^{2}}}{8}$

Step 6

Simplify the numerator.

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

$\frac{\sqrt{64 - 3^{2}}}{8}$

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

$\frac{\sqrt{64 - 1 \cdot 9}}{8}$

Multiply $- 1$ by $9$ .

$\frac{\sqrt{64 - 9}}{8}$

Subtract $9$ from $64$ .

$\frac{\sqrt{55}}{8}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{55}}{8}$

Decimal Form:

$0.92702481 \dots$

Step 8

Find the Eccentricity ((x+1)^2)/9+((y-5)^2)/64=1

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