Graph (x^2)/16+(y^2)/64=1

$\frac{x^{2}}{16} + \frac{y^{2}}{64} = 1$

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 $1$ .

$\frac{x^{2}}{16} + \frac{y^{2}}{64} = 1$

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.

$a = 8$

$b = 4$

$k = 0$

$h = 0$

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.

$\sqrt{{(8)}^{2} - {(4)}^{2}}$

Simplify.

Raise $8$ to the power of $2$ .

$\sqrt{64 - {(4)}^{2}}$

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

$\sqrt{64 - 1 \cdot 16}$

Multiply $- 1$ by $16$ .

$\sqrt{64 - 16}$

Subtract $16$ from $64$ .

$\sqrt{48}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$\sqrt{16 (3)}$

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

$\sqrt{4^{2} \cdot 3}$

Pull terms out from under the radical.

$4 \sqrt{3}$

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.

$(0, 0 + 8)$

Simplify.

$(0, 8)$

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.

$(0, 0 - (8))$

Simplify.

$(0, - 8)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(0, 8)$

${Vertex}_{2}$ : $(0, - 8)$

${Vertex}_{1}$ : $(0, 8)$

${Vertex}_{2}$ : $(0, - 8)$

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.

$(0, 0 + 4 \sqrt{3})$

Simplify.

$(0, 4 \sqrt{3})$

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.

$(0, 0 - (4 \sqrt{3}))$

Simplify.

$(0, - 4 \sqrt{3})$

Ellipses have two foci.

${Focus}_{1}$ : $(0, 4 \sqrt{3})$

${Focus}_{2}$ : $(0, - 4 \sqrt{3})$

${Focus}_{1}$ : $(0, 4 \sqrt{3})$

${Focus}_{2}$ : $(0, - 4 \sqrt{3})$

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.

$\frac{\sqrt{{(8)}^{2} - {(4)}^{2}}}{8}$

Simplify.

Simplify the numerator.

Raise $8$ to the power of $2$ .

$\frac{\sqrt{64 - 4^{2}}}{8}$

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

$\frac{\sqrt{64 - 1 \cdot 16}}{8}$

Multiply $- 1$ by $16$ .

$\frac{\sqrt{64 - 16}}{8}$

Subtract $16$ from $64$ .

$\frac{\sqrt{48}}{8}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$\frac{\sqrt{16 (3)}}{8}$

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

$\frac{\sqrt{4^{2} \cdot 3}}{8}$

Pull terms out from under the radical.

$\frac{4 \sqrt{3}}{8}$

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4 \sqrt{3}$ .

$\frac{4 (\sqrt{3})}{8}$

Cancel the common factors.

Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 9

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

Center: $(0, 0)$

${Vertex}_{1}$ : $(0, 8)$

${Vertex}_{2}$ : $(0, - 8)$

${Focus}_{1}$ : $(0, 4 \sqrt{3})$

${Focus}_{2}$ : $(0, - 4 \sqrt{3})$

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

Step 10

Graph (x^2)/16+(y^2)/64=1

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