Find the Asymptotes (x^2)/8+(y^2)/12=1

$\frac{x^{2}}{8} + \frac{y^{2}}{12} = 1$

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 $1$ .

$\frac{x^{2}}{8} + \frac{y^{2}}{12} = 1$

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.

$a = 2 \sqrt{3}$

$b = 2 \sqrt{2}$

$k = 0$

$h = 0$

Step 4

The center of an ellipse follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(0, 0)$

Step 5

Find $c$ , the distance from the center to a focus.

Find the distance from the center to a focus of the ellipse by using the following formula.

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

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

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

Simplify.

Simplify the expression.

Apply the product rule to $2 \sqrt{3}$ .

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

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

$\sqrt{4 {\sqrt{3}}^{2} - {(2 \sqrt{2})}^{2}}$

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

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

$\sqrt{4 {(3^{\frac{1}{2}})}^{2} - {(2 \sqrt{2})}^{2}}$

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

$\sqrt{4 \cdot 3^{\frac{1}{2} \cdot 2} - {(2 \sqrt{2})}^{2}}$

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

$\sqrt{4 \cdot 3^{\frac{2}{2}} - {(2 \sqrt{2})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{4 \cdot 3^{\frac{2}{2}} - {(2 \sqrt{2})}^{2}}$

Rewrite the expression.

$\sqrt{4 \cdot 3^{1} - {(2 \sqrt{2})}^{2}}$

Evaluate the exponent.

$\sqrt{4 \cdot 3 - {(2 \sqrt{2})}^{2}}$

Simplify the expression.

Multiply $4$ by $3$ .

$\sqrt{12 - {(2 \sqrt{2})}^{2}}$

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

$\sqrt{12 - (2^{2} {\sqrt{2}}^{2})}$

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

$\sqrt{12 - (4 {\sqrt{2}}^{2})}$

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

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

$\sqrt{12 - (4 {(2^{\frac{1}{2}})}^{2})}$

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

$\sqrt{12 - (4 \cdot 2^{\frac{1}{2} \cdot 2})}$

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

$\sqrt{12 - (4 \cdot 2^{\frac{2}{2}})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{12 - (4 \cdot 2^{\frac{2}{2}})}$

Rewrite the expression.

$\sqrt{12 - (4 \cdot 2^{1})}$

Evaluate the exponent.

$\sqrt{12 - (4 \cdot 2)}$

Multiply $- (4 \cdot 2)$ .

Multiply $4$ by $2$ .

$\sqrt{12 - 1 \cdot 8}$

Multiply $- 1$ by $8$ .

$\sqrt{12 - 8}$

Subtract $8$ from $12$ .

$\sqrt{4}$

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

$\sqrt{2^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$2$

Step 6

Find the vertices.

The first vertex of an ellipse can be found by adding $a$ to $k$ .

$(h, k + a)$

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(0, 0 + 2 \sqrt{3})$

Simplify.

$(0, 2 \sqrt{3})$

The second vertex of an ellipse can be found by subtracting $a$ from $k$ .

$(h, k - a)$

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(0, 0 - (2 \sqrt{3}))$

Simplify.

$(0, - 2 \sqrt{3})$

Ellipses have two vertices.

${Vertex}_{1}$ : $(0, 2 \sqrt{3})$

${Vertex}_{2}$ : $(0, - 2 \sqrt{3})$

${Vertex}_{1}$ : $(0, 2 \sqrt{3})$

${Vertex}_{2}$ : $(0, - 2 \sqrt{3})$

Step 7

Find the foci.

The first focus of an ellipse can be found by adding $c$ to $k$ .

$(h, k + c)$

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(0, 0 + 2)$

Simplify.

$(0, 2)$

The first focus of an ellipse can be found by subtracting $c$ from $k$ .

$(h, k - c)$

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(0, 0 - (2))$

Simplify.

$(0, - 2)$

Ellipses have two foci.

${Focus}_{1}$ : $(0, 2)$

${Focus}_{2}$ : $(0, - 2)$

${Focus}_{1}$ : $(0, 2)$

${Focus}_{2}$ : $(0, - 2)$

Step 8

Find the eccentricity.

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.

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

Simplify.

Simplify the numerator.

Apply the product rule to $2 \sqrt{3}$ .

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

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

$\frac{\sqrt{4 {\sqrt{3}}^{2} - {(2 \sqrt{2})}^{2}}}{2 \sqrt{3}}$

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

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

$\frac{\sqrt{4 {(3^{\frac{1}{2}})}^{2} - {(2 \sqrt{2})}^{2}}}{2 \sqrt{3}}$

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

$\frac{\sqrt{4 \cdot 3^{\frac{1}{2} \cdot 2} - {(2 \sqrt{2})}^{2}}}{2 \sqrt{3}}$

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

$\frac{\sqrt{4 \cdot 3^{\frac{2}{2}} - {(2 \sqrt{2})}^{2}}}{2 \sqrt{3}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{4 \cdot 3^{\frac{2}{2}} - {(2 \sqrt{2})}^{2}}}{2 \sqrt{3}}$

Rewrite the expression.

$\frac{\sqrt{4 \cdot 3^{1} - {(2 \sqrt{2})}^{2}}}{2 \sqrt{3}}$

Evaluate the exponent.

$\frac{\sqrt{4 \cdot 3 - {(2 \sqrt{2})}^{2}}}{2 \sqrt{3}}$

Multiply $4$ by $3$ .

$\frac{\sqrt{12 - {(2 \sqrt{2})}^{2}}}{2 \sqrt{3}}$

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

$\frac{\sqrt{12 - (2^{2} {\sqrt{2}}^{2})}}{2 \sqrt{3}}$

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

$\frac{\sqrt{12 - (4 {\sqrt{2}}^{2})}}{2 \sqrt{3}}$

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

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

$\frac{\sqrt{12 - (4 {(2^{\frac{1}{2}})}^{2})}}{2 \sqrt{3}}$

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

$\frac{\sqrt{12 - (4 \cdot 2^{\frac{1}{2} \cdot 2})}}{2 \sqrt{3}}$

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

$\frac{\sqrt{12 - (4 \cdot 2^{\frac{2}{2}})}}{2 \sqrt{3}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{12 - (4 \cdot 2^{\frac{2}{2}})}}{2 \sqrt{3}}$

Rewrite the expression.

$\frac{\sqrt{12 - (4 \cdot 2^{1})}}{2 \sqrt{3}}$

Evaluate the exponent.

$\frac{\sqrt{12 - (4 \cdot 2)}}{2 \sqrt{3}}$

Multiply $- (4 \cdot 2)$ .

Multiply $4$ by $2$ .

$\frac{\sqrt{12 - 1 \cdot 8}}{2 \sqrt{3}}$

Multiply $- 1$ by $8$ .

$\frac{\sqrt{12 - 8}}{2 \sqrt{3}}$

Subtract $8$ from $12$ .

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

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

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

Pull terms out from under the radical, assuming positive real numbers.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 9

These values represent the important values for graphing and analyzing an ellipse.

Center: $(0, 0)$

${Vertex}_{1}$ : $(0, 2 \sqrt{3})$

${Vertex}_{2}$ : $(0, - 2 \sqrt{3})$

${Focus}_{1}$ : $(0, 2)$

${Focus}_{2}$ : $(0, - 2)$

Eccentricity: $\frac{\sqrt{3}}{3}$

Step 10

Find the Asymptotes (x^2)/8+(y^2)/12=1

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