Find the Eccentricity (x^2)/25+(y^2)/4=1

$\frac{x^{2}}{25} + \frac{y^{2}}{4} = 1$

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 $1$ .

$\frac{x^{2}}{25} + \frac{y^{2}}{4} = 1$

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.

$a = 5$

$b = 2$

$k = 0$

$h = 0$

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.

$\frac{\sqrt{{(5)}^{2} - {(2)}^{2}}}{5}$

Step 6

Simplify the numerator.

Raise $5$ to the power of $2$ .

$\frac{\sqrt{25 - 2^{2}}}{5}$

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

$\frac{\sqrt{25 - 1 \cdot 4}}{5}$

Multiply $- 1$ by $4$ .

$\frac{\sqrt{25 - 4}}{5}$

Subtract $4$ from $25$ .

$\frac{\sqrt{21}}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{21}}{5}$

Decimal Form:

$0.91651513 \dots$

Step 8

Find the Eccentricity (x^2)/25+(y^2)/4=1

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