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If alpha and beta are the eccentric angles of the endpoints of a focal chord of a standard ellipse, find the eccentricity of the ellipse in terms of alpha and beta.

1. (sin(alpha) + sin(beta))/sin(alpha + beta)
2. (cos(alpha) + cos(beta))/cos(alpha + beta)
3. (sin(alpha) - sin(beta))/sin(alpha - beta)
4. (cos(alpha) - cos(beta))/cos(alpha - beta)

Correct answer: (sin(alpha) + sin(beta))/sin(alpha + beta)

Solution

Substituting the focus (ae, 0) into the chord-of-an-ellipse equation gives e = cos((a-b)/2)/cos((a+b)/2), which simplifies to (sin alpha + sin beta)/sin(alpha + beta).

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