Simplify each term in the equation in order to set the right side equal to <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> . The standard form of an ellipse or hyperbola requires the right side of the equation be <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

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.

Match the values in this ellipse to those of the standard form. The variable <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> represents the radius of the major axis of the ellipse, <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> represents the radius of the minor axis of the ellipse, <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> represents the x-offset from the origin, and <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> represents the y-offset from the origin.

The center of an ellipse follows the form of <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mstyle></math> . Substitute in the values of <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> .

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

Substitute the values of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> in the formula.

Simplify.

Raise <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> .

The first vertex of an ellipse can be found by adding <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> to <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> .

Substitute the known values of <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> , <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> , and <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> into the formula.

Simplify.

The second vertex of an ellipse can be found by subtracting <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> from <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> .

Substitute the known values of <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> , <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> , and <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> into the formula.

Simplify.

Ellipses have two vertices.

The first focus of an ellipse can be found by adding <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> to <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> .

Substitute the known values of <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> , <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> , and <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> into the formula.

The second vertex of an ellipse can be found by subtracting <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> from <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> .

Substitute the known values of <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> , <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> , and <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> into the formula.

Simplify.

Ellipses have two foci.

Find the eccentricity by using the following formula.

Substitute the values of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> into the formula.

Simplify the numerator.

Raise <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> .

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

Center: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow></mstyle></math>

Eccentricity: <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>5</mn></msqrt></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math>

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