Find the Foci (x^2)/16+(y^2)/32=1

$\frac{x^{2}}{16} + \frac{y^{2}}{32} = 1$

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 $1$ .

$\frac{x^{2}}{16} + \frac{y^{2}}{32} = 1$

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.

$a = 4 \sqrt{2}$

$b = 4$

$k = 0$

$h = 0$

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.

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

Simplify.

Simplify the expression.

Apply the product rule to $4 \sqrt{2}$ .

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

Raise $4$ to the power of $2$ .

$\sqrt{16 {\sqrt{2}}^{2} - {(4)}^{2}}$

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

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

$\sqrt{16 {(2^{\frac{1}{2}})}^{2} - {(4)}^{2}}$

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

$\sqrt{16 \cdot 2^{\frac{1}{2} \cdot 2} - {(4)}^{2}}$

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

$\sqrt{16 \cdot 2^{\frac{2}{2}} - {(4)}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{16 \cdot 2^{\frac{2}{2}} - {(4)}^{2}}$

Rewrite the expression.

$\sqrt{16 \cdot 2^{1} - {(4)}^{2}}$

Evaluate the exponent.

$\sqrt{16 \cdot 2 - {(4)}^{2}}$

Simplify the expression.

Multiply $16$ by $2$ .

$\sqrt{32 - {(4)}^{2}}$

Raise $4$ to the power of $2$ .

$\sqrt{32 - 1 \cdot 16}$

Multiply $- 1$ by $16$ .

$\sqrt{32 - 16}$

Subtract $16$ from $32$ .

$\sqrt{16}$

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2}}$

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

$4$

Step 5

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.

$(0, 0 + 4)$

Simplify.

$(0, 4)$

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.

$(0, 0 - (4))$

Simplify.

$(0, - 4)$

Ellipses have two foci.

${Focus}_{1}$ : $(0, 4)$

${Focus}_{2}$ : $(0, - 4)$

${Focus}_{1}$ : $(0, 4)$

${Focus}_{2}$ : $(0, - 4)$

Step 6

Find the Foci (x^2)/16+(y^2)/32=1

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