Find the Eccentricity ((x-5)^2)/52+((y+1)^2)/64=1

$\frac{{(x - 5)}^{2}}{52} + \frac{{(y + 1)}^{2}}{64} = 1$

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 $1$ .

$\frac{{(x - 5)}^{2}}{52} + \frac{{(y + 1)}^{2}}{64} = 1$

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.

$a = 8$

$b = 2 \sqrt{13}$

$k = - 1$

$h = 5$

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.

$\frac{\sqrt{{(8)}^{2} - {(2 \sqrt{13})}^{2}}}{8}$

Step 6

Simplify.

Simplify the numerator.

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

$\frac{\sqrt{64 - {(2 \sqrt{13})}^{2}}}{8}$

Apply the product rule to $2 \sqrt{13}$ .

$\frac{\sqrt{64 - (2^{2} {\sqrt{13}}^{2})}}{8}$

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

$\frac{\sqrt{64 - (4 {\sqrt{13}}^{2})}}{8}$

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{\sqrt{64 - (4 {(13^{\frac{1}{2}})}^{2})}}{8}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{64 - (4 \cdot 13^{\frac{1}{2} \cdot 2})}}{8}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{64 - (4 \cdot 13^{\frac{2}{2}})}}{8}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{64 - (4 \cdot 13^{\frac{2}{2}})}}{8}$

Rewrite the expression.

$\frac{\sqrt{64 - (4 \cdot 13^{1})}}{8}$

Evaluate the exponent.

$\frac{\sqrt{64 - (4 \cdot 13)}}{8}$

Multiply $- (4 \cdot 13)$ .

Multiply $4$ by $13$ .

$\frac{\sqrt{64 - 1 \cdot 52}}{8}$

Multiply $- 1$ by $52$ .

$\frac{\sqrt{64 - 52}}{8}$

Subtract $52$ from $64$ .

$\frac{\sqrt{12}}{8}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)}}{8}$

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 3}}{8}$

Pull terms out from under the radical.

$\frac{2 \sqrt{3}}{8}$

Cancel the common factor of $2$ and $8$ .

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

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

Cancel the common factors.

Factor $2$ out of $8$ .

$\frac{2 \sqrt{3}}{2 \cdot 4}$

Cancel the common factor.

$\frac{2 \sqrt{3}}{2 \cdot 4}$

Rewrite the expression.

$\frac{\sqrt{3}}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{4}$

Decimal Form:

$0.43301270 \dots$

Step 8

Find the Eccentricity ((x-5)^2)/52+((y+1)^2)/64=1

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