Ellipse Math Example 2

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Example 2

medium
Find the foci of the ellipse x216+y225=1\frac{x^2}{16} + \frac{y^2}{25} = 1.

Solution

  1. 1
    Identify a2=25a^2 = 25 (under y2y^2, since 25>1625 > 16) and b2=16b^2 = 16. The major axis is vertical.
  2. 2
    Find cc using c2=a2−b2=25−16=9c^2 = a^2 - b^2 = 25 - 16 = 9, so c=3c = 3.
  3. 3
    Since the major axis is along the yy-axis, the foci are at (0,±c)=(0,±3)(0, \pm c) = (0, \pm 3).

Answer

Foci at (0,3) and (0,−3)\text{Foci at } (0, 3) \text{ and } (0, -3)
For an ellipse, the foci lie on the major axis at distance c=a2−b2c = \sqrt{a^2 - b^2} from the center. The relationship c2=a2−b2c^2 = a^2 - b^2 comes from the definition: the sum of distances from any point on the ellipse to the two foci equals 2a2a.

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: (x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1.

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More Ellipse Examples

Example 1 easy

Find the lengths of the semi-major and semi-minor axes 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

Example 4 hard

Find the eccentricity of the ellipse [formula] and describe what it tells us about the shape.

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