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

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

Find the standard form of the ellipse.

Divide each term by $4$ to make the right side equal to one.

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

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 $1$ .

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

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.

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

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.

$a = 2$

$b = 2$

$k = 2$

$h = 1$

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

$(1, 2)$

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.

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

Simplify.

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

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

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

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

Multiply $- 1$ by $4$ .

$\sqrt{4 - 4}$

Subtract $4$ from $4$ .

$\sqrt{0}$

Rewrite $0$ as $0^{2}$ .

$\sqrt{0^{2}}$

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

$0$

Find the vertices.

The first vertex of an ellipse can be found by adding $a$ to $h$ .

$(h + a, k)$

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(1 + 2, 2)$

Simplify.

$(3, 2)$

The second vertex of an ellipse can be found by subtracting $a$ from $h$ .

$(h - a, k)$

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(1 - (2), 2)$

Simplify.

$(- 1, 2)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(3, 2)$

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

${Vertex}_{1}$ : $(3, 2)$

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

Find the foci.

The first focus of an ellipse can be found by adding $c$ to $h$ .

$(h + c, k)$

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(1 + 0, 2)$

Simplify.

$(1, 2)$

The second focus of an ellipse can be found by subtracting $c$ from $h$ .

$(h - c, k)$

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(1 - (0), 2)$

Simplify.

$(1, 2)$

Ellipses have two foci.

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

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

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

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

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.

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

Simplify.

Simplify the numerator.

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

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

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

$\frac{\sqrt{4 - 1 \cdot 4}}{2}$

Multiply $- 1$ by $4$ .

$\frac{\sqrt{4 - 4}}{2}$

Subtract $4$ from $4$ .

$\frac{\sqrt{0}}{2}$

Rewrite $0$ as $0^{2}$ .

$\frac{\sqrt{0^{2}}}{2}$

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

$\frac{0}{2}$

Divide $0$ by $2$ .

$0$

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

Center: $(1, 2)$

${Vertex}_{1}$ : $(3, 2)$

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

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

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

Eccentricity: $0$

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

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