Graph ((x-2)^2)/4+((y-4)^2)/9=1

$\frac{{(x - 2)}^{2}}{4} + \frac{{(y - 4)}^{2}}{9} = 1$

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 $1$ .

$\frac{{(x - 2)}^{2}}{4} + \frac{{(y - 4)}^{2}}{9} = 1$

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.

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

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.

$a = 3$

$b = 2$

$k = 4$

$h = 2$

Step 4

The center of an ellipse follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(2, 4)$

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.

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

Simplify.

Raise $3$ to the power of $2$ .

$\sqrt{9 - {(2)}^{2}}$

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

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

Multiply $- 1$ by $4$ .

$\sqrt{9 - 4}$

Subtract $4$ from $9$ .

$\sqrt{5}$

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.

$(2, 4 + 3)$

Simplify.

$(2, 7)$

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.

$(2, 4 - (3))$

Simplify.

$(2, 1)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(2, 7)$

${Vertex}_{2}$ : $(2, 1)$

${Vertex}_{1}$ : $(2, 7)$

${Vertex}_{2}$ : $(2, 1)$

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.

$(2, 4 + \sqrt{5})$

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.

$(2, 4 - (\sqrt{5}))$

Simplify.

$(2, 4 - \sqrt{5})$

Ellipses have two foci.

${Focus}_{1}$ : $(2, 4 + \sqrt{5})$

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

${Focus}_{1}$ : $(2, 4 + \sqrt{5})$

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

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.

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

Simplify the numerator.

Raise $3$ to the power of $2$ .

$\frac{\sqrt{9 - 2^{2}}}{3}$

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

$\frac{\sqrt{9 - 1 \cdot 4}}{3}$

Multiply $- 1$ by $4$ .

$\frac{\sqrt{9 - 4}}{3}$

Subtract $4$ from $9$ .

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

Step 9

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

Center: $(2, 4)$

${Vertex}_{1}$ : $(2, 7)$

${Vertex}_{2}$ : $(2, 1)$

${Focus}_{1}$ : $(2, 4 + \sqrt{5})$

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

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

Step 10

Graph ((x-2)^2)/4+((y-4)^2)/9=1

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