Algebra Examples

\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

Step 1

Simplify each term in the equation in order to set the right side equal to

1

$1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be

1

$1$ .

\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

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

$\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1$

Step 3

Match the values in this ellipse to those of the standard form. The variable

a

$a$ represents the radius of the major axis of the ellipse,

b

$b$ represents the radius of the minor axis of the ellipse,

h

$h$ represents the x-offset from the origin, and

k

$k$ represents the y-offset from the origin.

a = 4

$a = 4$

b = 3

$b = 3$

k = 0

$k = 0$

h = 0

$h = 0$

Step 4

The center of an ellipse follows the form of

(h, k)

$(h, k)$ . Substitute in the values of

h

$h$ and

k

$k$ .

(0, 0)

$(0, 0)$

Step 5

Find

c

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

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Step 5.1

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

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

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

Step 5.2

Substitute the values of

a

$a$ and

b

$b$ in the formula.

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

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

Step 5.3

Simplify.

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Step 5.3.1

Raise

4

$4$ to the power of

2

$2$ .

\sqrt{16 - {(3)}^{2}}

$\sqrt{16 - {(3)}^{2}}$

Step 5.3.2

Raise

3

$3$ to the power of

2

$2$ .

\sqrt{16 - 1 \cdot 9}

$\sqrt{16 - 1 \cdot 9}$

Step 5.3.3

Multiply

- 1

$- 1$ by

9

$9$ .

\sqrt{16 - 9}

$\sqrt{16 - 9}$

Step 5.3.4

Subtract

9

$9$ from

16

$16$ .

\sqrt{7}

$\sqrt{7}$

\sqrt{7}

$\sqrt{7}$

\sqrt{7}

$\sqrt{7}$

Step 6

Find the vertices.

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Step 6.1

The first vertex of an ellipse can be found by adding

a

$a$ to

h

$h$ .

(h + a, k)

$(h + a, k)$

Step 6.2

Substitute the known values of

h

$h$ ,

a

$a$ , and

k

$k$ into the formula.

(0 + 4, 0)

$(0 + 4, 0)$

Step 6.3

Simplify.

(4, 0)

$(4, 0)$

Step 6.4

The second vertex of an ellipse can be found by subtracting

a

$a$ from

h

$h$ .

(h - a, k)

$(h - a, k)$

Step 6.5

Substitute the known values of

h

$h$ ,

a

$a$ , and

k

$k$ into the formula.

(0 - (4), 0)

$(0 - (4), 0)$

Step 6.6

Simplify.

(- 4, 0)

$(- 4, 0)$

Step 6.7

Ellipses have two vertices.

{Vertex}_{1}

${Vertex}_{1}$ :

(4, 0)

$(4, 0)$

{Vertex}_{2}

${Vertex}_{2}$ :

(- 4, 0)

$(- 4, 0)$

{Vertex}_{1}

${Vertex}_{1}$ :

(4, 0)

$(4, 0)$

{Vertex}_{2}

${Vertex}_{2}$ :

(- 4, 0)

$(- 4, 0)$

Step 7

Find the foci.

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Step 7.1

The first focus of an ellipse can be found by adding

c

$c$ to

h

$h$ .

(h + c, k)

$(h + c, k)$

Step 7.2

Substitute the known values of

h

$h$ ,

c

$c$ , and

k

$k$ into the formula.

(0 + \sqrt{7}, 0)

$(0 + \sqrt{7}, 0)$

Step 7.3

Simplify.

(\sqrt{7}, 0)

$(\sqrt{7}, 0)$

Step 7.4

The second focus of an ellipse can be found by subtracting

c

$c$ from

h

$h$ .

(h - c, k)

$(h - c, k)$

Step 7.5

Substitute the known values of

h

$h$ ,

c

$c$ , and

k

$k$ into the formula.

(0 - (\sqrt{7}), 0)

$(0 - (\sqrt{7}), 0)$

Step 7.6

Simplify.

(- \sqrt{7}, 0)

$(- \sqrt{7}, 0)$

Step 7.7

Ellipses have two foci.

{Focus}_{1}

${Focus}_{1}$ :

(\sqrt{7}, 0)

$(\sqrt{7}, 0)$

{Focus}_{2}

${Focus}_{2}$ :

(- \sqrt{7}, 0)

$(- \sqrt{7}, 0)$

{Focus}_{1}

${Focus}_{1}$ :

(\sqrt{7}, 0)

$(\sqrt{7}, 0)$

{Focus}_{2}

${Focus}_{2}$ :

(- \sqrt{7}, 0)

$(- \sqrt{7}, 0)$

Step 8

Find the eccentricity.

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Step 8.1

Find the eccentricity by using the following formula.

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

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

Step 8.2

Substitute the values of

a

$a$ and

b

$b$ into the formula.

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

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

Step 8.3

Simplify the numerator.

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Step 8.3.1

Raise

$4$ to the power of

$2$ .

$\frac{\sqrt{16 - 3^{2}}}{4}$

Step 8.3.2

Raise

$3$ to the power of

$2$ .

$\frac{\sqrt{16 - 1 \cdot 9}}{4}$

Step 8.3.3

Multiply

$- 1$ by

$9$ .

$\frac{\sqrt{16 - 9}}{4}$

Step 8.3.4

Subtract

$9$ from

$16$ .

$\frac{\sqrt{7}}{4}$

Step 9

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

Center:

$(0, 0)$

${Vertex}_{1}$ :

$(4, 0)$

${Vertex}_{2}$ :

$(- 4, 0)$

${Focus}_{1}$ :

$(\sqrt{7}, 0)$

${Focus}_{2}$ :

$(- \sqrt{7}, 0)$

Eccentricity:

$\frac{\sqrt{7}}{4}$

Step 10