Find the Foci 3x^2+8y^2=24

$3 x^{2} + 8 y^{2} = 24$

Step 1

Find the standard form of the ellipse.

Divide each term by $24$ to make the right side equal to one.

$\frac{3 x^{2}}{24} + \frac{8 y^{2}}{24} = \frac{24}{24}$

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 $1$ .

$\frac{x^{2}}{8} + \frac{y^{2}}{3} = 1$

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.

$a = 2 \sqrt{2}$

$b = \sqrt{3}$

$k = 0$

$h = 0$

Step 4

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.

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

Simplify.

Simplify the expression.

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

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $4$ by $2$ .

$\sqrt{8 - {(\sqrt{3})}^{2}}$

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

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

$\sqrt{8 - {(3^{\frac{1}{2}})}^{2}}$

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\sqrt{8 - 3^{1}}$

Evaluate the exponent.

$\sqrt{8 - 1 \cdot 3}$

Simplify the expression.

Multiply $- 1$ by $3$ .

$\sqrt{8 - 3}$

Subtract $3$ from $8$ .

$\sqrt{5}$

Step 5

Find the foci.

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

$(h + c, k)$

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

$(0 + \sqrt{5}, 0)$

Simplify.

$(\sqrt{5}, 0)$

The second focus of an ellipse can be found by subtracting $c$ from $h$ .

$(h - c, k)$

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

$(0 - (\sqrt{5}), 0)$

Simplify.

$(- \sqrt{5}, 0)$

Ellipses have two foci.

${Focus}_{1}$ : $(\sqrt{5}, 0)$

${Focus}_{2}$ : $(- \sqrt{5}, 0)$

${Focus}_{1}$ : $(\sqrt{5}, 0)$

${Focus}_{2}$ : $(- \sqrt{5}, 0)$

Step 6

Find the Foci 3x^2+8y^2=24

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