IXL - Find the eccentricity of an ellipse (Precalculus practice)

The graph shows an ellipse. What is its eccentricity?

Write your answer in simplified, rationalized form.

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question

The graph shows an ellipse. What is its eccentricity?

Write your answer in simplified, rationalized form.

key idea

The equation of an ellipse that has a vertical orientation in standard form is

(

–

)

(

–

)

where

is the distance from the center to a vertex, and

is the distance from the center to a co-vertex.

The eccentricity is

where

is the distance from the center to a focus, and

–

The eccentricity is in the interval [0, 1) and measures how close an ellipse is to being a circle. An eccentricity of 0 means the ellipse is a circle. An eccentricity close to 1 means the ellipse is far from being a circle and is close to being a parabola.

solution

The eccentricity cannot be found by simply looking at a graph. So, first find the center. Then, find

and

Finally, calculate the length

and find

Find the center.

Look at the graph of the ellipse:

The center is

(

)

Find

and

The ellipse is taller than it is wide, so it has a vertical orientation. This means the major axis is vertical and the minor axis is horizontal. To find a, count the units from the center to the ellipse along the major axis. To find b, count the units from the center to the ellipse along the minor axis.

So,

and

Calculate

then take the square root.

–

Since

is a distance, it is positive. So

Find

The eccentricity is:

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U.7Find properties of ellipses

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