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Ellipse

See how to solve an ellipse
(major/minor axis, foci, equation).
5 examples and their solutions.

Definition



PF + PF' = (constant)
An ellipse is the set of points that satisfy
PF + PF' = (constant, major axis).
F, F': foci

Ellipse: x2a2 + y2b2 = 1 (a > b)

Equation

x2a2 + y2b2 = 1 (a > b)



a2 - b2 = c2

Major Axis: 2a
Minor Axis: 2b
Foci: (c, 0), (-c, 0)
The major axis is the longest diameter: 2a.
The minor axis is the shortest diameter: 2b.
a > b
Then the foci are located horizontally.
By using a2 - b2 = c2,
you can find the foci (±c, 0).

Example

x225 + y216 = 1
1. Major axis?
2. Minor axis?
3. Foci?
Solution

Example

Foci: (4, 0), (-4, 0)
Major axis: 10
Equation of the ellipse?
Solution

Example

Foci: (0, 1), (4, 1)
Major axis: 6
Equation of the ellipse?
Solution

Ellipse: x2a2 + y2b2 = 1 (a < b)

Equation

x2a2 + y2b2 = 1 (a < b)



b2 - a2 = c2

Major Axis: 2b
Minor Axis: 2a
Foci: (0, c), (0, -c)

Example

9x2+ 4y2 = 36
1. Major axis?
2. Minor axis?
3. Foci?
Solution

Example

Foci: (0, 2), (0, -2)
Major axis: 8
Equation of the ellipse?
Solution