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P is any point on an ellipse with foci S1 and S2 and eccentricity 1/2. Let angle PS1S2 = alpha, angle PS2S1 = beta, and angle S1PS2 = gamma. Then cot(alpha/2), cot(gamma/2), cot(beta/2) are in which progression?

1. A.P.
2. G.P.
3. H.P.
4. NOT A.P., G.P. or H.P.

Correct answer: A.P.

Solution

Applying the half-angle cotangent formulas in the focal triangle and using r1+r2 = 2a, S1S2 = 2ae shows the three cotangents form an arithmetic progression.

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