Find the Eccentricity x^2+3y^2=15

$x^{2} + 3 y^{2} = 15$

Step 1

Divide each term by $15$ to make the right side equal to one.

$\frac{x^{2}}{15} + \frac{3 y^{2}}{15} = \frac{15}{15}$

Step 2

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 $1$ .

$\frac{x^{2}}{15} + \frac{y^{2}}{5} = 1$

Step 3

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 4

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.

$a = \sqrt{15}$

$b = \sqrt{5}$

$k = 0$

$h = 0$

Step 5

Find the eccentricity by using the following formula.

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

Step 6

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

$\frac{\sqrt{{(\sqrt{15})}^{2} - {(\sqrt{5})}^{2}}}{\sqrt{15}}$

Step 7

Simplify.

Simplify the numerator.

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\frac{\sqrt{{(15^{\frac{1}{2}})}^{2} - {\sqrt{5}}^{2}}}{\sqrt{15}}$

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

$\frac{\sqrt{15^{\frac{1}{2} \cdot 2} - {\sqrt{5}}^{2}}}{\sqrt{15}}$

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

$\frac{\sqrt{15^{\frac{2}{2}} - {\sqrt{5}}^{2}}}{\sqrt{15}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{15^{\frac{2}{2}} - {\sqrt{5}}^{2}}}{\sqrt{15}}$

Rewrite the expression.

$\frac{\sqrt{15^{1} - {\sqrt{5}}^{2}}}{\sqrt{15}}$

Evaluate the exponent.

$\frac{\sqrt{15 - {\sqrt{5}}^{2}}}{\sqrt{15}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt{15 - {(5^{\frac{1}{2}})}^{2}}}{\sqrt{15}}$

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

$\frac{\sqrt{15 - 5^{\frac{1}{2} \cdot 2}}}{\sqrt{15}}$

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

$\frac{\sqrt{15 - 5^{\frac{2}{2}}}}{\sqrt{15}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{15 - 5^{\frac{2}{2}}}}{\sqrt{15}}$

Rewrite the expression.

$\frac{\sqrt{15 - 5^{1}}}{\sqrt{15}}$

Evaluate the exponent.

$\frac{\sqrt{15 - 1 \cdot 5}}{\sqrt{15}}$

Multiply $- 1$ by $5$ .

$\frac{\sqrt{15 - 5}}{\sqrt{15}}$

Subtract $5$ from $15$ .

$\frac{\sqrt{10}}{\sqrt{15}}$

Combine $\sqrt{10}$ and $\sqrt{15}$ into a single radical.

$\sqrt{\frac{10}{15}}$

Cancel the common factor of $10$ and $15$ .

Factor $5$ out of $10$ .

$\sqrt{\frac{5 (2)}{15}}$

Cancel the common factors.

Factor $5$ out of $15$ .

$\sqrt{\frac{5 \cdot 2}{5 \cdot 3}}$

Cancel the common factor.

$\sqrt{\frac{5 \cdot 2}{5 \cdot 3}}$

Rewrite the expression.

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

Rewrite $\sqrt{\frac{2}{3}}$ as $\frac{\sqrt{2}}{\sqrt{3}}$ .

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

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{\sqrt{2} \sqrt{3}}{3^{1}}$

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.81649658 \dots$

Step 9

Find the Eccentricity x^2+3y^2=15

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