Find the Asymptotes (y^2)/144+(x^2)/64=1

$\frac{y^{2}}{144} + \frac{x^{2}}{64} = 1$

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 $1$ .

$\frac{y^{2}}{144} + \frac{x^{2}}{64} = 1$

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.

$a = 12$

$b = 8$

$k = 0$

$h = 0$

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.

$\sqrt{{(12)}^{2} - {(8)}^{2}}$

Simplify.

Raise $12$ to the power of $2$ .

$\sqrt{144 - {(8)}^{2}}$

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

$\sqrt{144 - 1 \cdot 64}$

Multiply $- 1$ by $64$ .

$\sqrt{144 - 64}$

Subtract $64$ from $144$ .

$\sqrt{80}$

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

$\sqrt{16 (5)}$

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

$\sqrt{4^{2} \cdot 5}$

Pull terms out from under the radical.

$4 \sqrt{5}$

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.

$(0, 0 + 12)$

Simplify.

$(0, 12)$

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.

$(0, 0 - (12))$

Simplify.

$(0, - 12)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(0, 12)$

${Vertex}_{2}$ : $(0, - 12)$

${Vertex}_{1}$ : $(0, 12)$

${Vertex}_{2}$ : $(0, - 12)$

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.

$(0, 0 + 4 \sqrt{5})$

Simplify.

$(0, 4 \sqrt{5})$

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.

$(0, 0 - (4 \sqrt{5}))$

Simplify.

$(0, - 4 \sqrt{5})$

Ellipses have two foci.

${Focus}_{1}$ : $(0, 4 \sqrt{5})$

${Focus}_{2}$ : $(0, - 4 \sqrt{5})$

${Focus}_{1}$ : $(0, 4 \sqrt{5})$

${Focus}_{2}$ : $(0, - 4 \sqrt{5})$

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.

$\frac{\sqrt{{(12)}^{2} - {(8)}^{2}}}{12}$

Simplify.

Simplify the numerator.

Raise $12$ to the power of $2$ .

$\frac{\sqrt{144 - 8^{2}}}{12}$

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

$\frac{\sqrt{144 - 1 \cdot 64}}{12}$

Multiply $- 1$ by $64$ .

$\frac{\sqrt{144 - 64}}{12}$

Subtract $64$ from $144$ .

$\frac{\sqrt{80}}{12}$

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

$\frac{\sqrt{16 (5)}}{12}$

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

$\frac{\sqrt{4^{2} \cdot 5}}{12}$

Pull terms out from under the radical.

$\frac{4 \sqrt{5}}{12}$

Cancel the common factor of $4$ and $12$ .

Factor $4$ out of $4 \sqrt{5}$ .

$\frac{4 (\sqrt{5})}{12}$

Cancel the common factors.

Factor $4$ out of $12$ .

$\frac{4 \sqrt{5}}{4 \cdot 3}$

Cancel the common factor.

$\frac{4 \sqrt{5}}{4 \cdot 3}$

Rewrite the expression.

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

Step 9

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

Center: $(0, 0)$

${Vertex}_{1}$ : $(0, 12)$

${Vertex}_{2}$ : $(0, - 12)$

${Focus}_{1}$ : $(0, 4 \sqrt{5})$

${Focus}_{2}$ : $(0, - 4 \sqrt{5})$

Eccentricity: $\frac{\sqrt{5}}{3}$

Step 10

Find the Asymptotes (y^2)/144+(x^2)/64=1

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