Ellipse Math Example 1

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Example 1

easy
Find the lengths of the semi-major and semi-minor axes of the ellipse x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1.

Solution

  1. 1
    The standard form is x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 where a>b>0a > b > 0.
  2. 2
    Here a2=25a^2 = 25 and b2=9b^2 = 9, so a=5a = 5 and b=3b = 3.
  3. 3
    The semi-major axis has length a=5a = 5 (along the xx-axis) and the semi-minor axis has length b=3b = 3 (along the yy-axis).

Answer

a=5 (semi-major),b=3 (semi-minor)a = 5 \text{ (semi-major)}, \quad b = 3 \text{ (semi-minor)}
An ellipse x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 has semi-major axis aa (the larger denominator's square root) and semi-minor axis bb. The major axis lies along whichever variable has the larger denominator.

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 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

Example 4 hard

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

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