Find the Eccentricity 4x^2+y^2=36

$4 x^{2} + y^{2} = 36$

Divide each term by $36$ to make the right side equal to one.

$\frac{4 x^{2}}{36} + \frac{y^{2}}{36} = \frac{36}{36}$

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 $1$ .

$\frac{x^{2}}{9} + \frac{y^{2}}{36} = 1$

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.

$a = 6$

$b = 3$

$k = 0$

$h = 0$

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.

$\frac{\sqrt{{(6)}^{2} - {(3)}^{2}}}{6}$

Simplify.

Simplify the numerator.

Raise $6$ to the power of $2$ .

$\frac{\sqrt{36 - 3^{2}}}{6}$

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

$\frac{\sqrt{36 - 1 \cdot 9}}{6}$

Multiply $- 1$ by $9$ .

$\frac{\sqrt{36 - 9}}{6}$

Subtract $9$ from $36$ .

$\frac{\sqrt{27}}{6}$

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$\frac{\sqrt{9 (3)}}{6}$

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 3}}{6}$

Pull terms out from under the radical.

$\frac{3 \sqrt{3}}{6}$

Cancel the common factor of $3$ and $6$ .

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

$\frac{3 (\sqrt{3})}{6}$

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.86602540 \dots$

Find the Eccentricity 4x^2+y^2=36

Related Questions

Support LCMGCF.com by disabling your adblocker.