Ellipse Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard

Find the eccentricity of the ellipse

\frac{(x-2)^2}{36} + \frac{(y+1)^2}{20} = 1

and describe what it tells us about the shape.

Solution

1
$a^2 = 36$ , $b^2 = 20$ . $c^2 = a^2 - b^2 = 16$ , so $c = 4$ . Eccentricity $e = \frac{c}{a} = \frac{4}{6} = \frac{2}{3}$ .
2
Since $0 < e < 1$ and $e = \frac{2}{3}$ is moderately close to $1$ , the ellipse is noticeably elongated (more oval than circular). A circle has $e = 0$ .

Answer

e = \frac{2}{3} \approx 0.667

Eccentricity

e = \frac{c}{a}

measures how elongated an ellipse is. Values range from

0

(perfect circle) to just below

1

(very elongated). At

e = 2/3

, the ellipse is moderately stretched — the foci are

2/3

of the way from center to vertex.

About Ellipse

The set of all points in a plane where the sum of the distances to two fixed points (foci) is constant. Standard form: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ .

Learn more about Ellipse →

More Ellipse Examples

Example 1 easy

Find the lengths of the semi-major and semi-minor axes of the ellipse [formula].

Example 2 medium

Find the foci of the ellipse [formula].

Example 3 medium

Write the equation of an ellipse centered at the origin with foci at [formula] and a major axis of l

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Equation of a Circle Conic Sections Overview Hyperbola