Ellipse Math Example 3

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

Example 3

medium
Write the equation of an ellipse centered at the origin with foci at (±4,0)(\pm 4, 0) and a major axis of length 1010.

Solution

  1. 1
    Major axis length =2a=10= 2a = 10, so a=5a = 5. Foci at (±4,0)(\pm 4, 0) give c=4c = 4. The major axis is horizontal.
  2. 2
    b2=a2−c2=25−16=9b^2 = a^2 - c^2 = 25 - 16 = 9. The equation is x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1.

Answer

x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1
Given the foci and major axis length, use cc (distance from center to focus) and aa (semi-major axis) to find bb via b2=a2−c2b^2 = a^2 - c^2. The foci lie along the direction of the major axis.

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.

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 4 hard

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

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