Multiple choice

The distances of three points on a pole from its foot are in A.P. If the angles of elevations of these points from a point on the ground are $\alpha, \beta$ and $\gamma$, then $\cot\beta \cot \gamma, \cot \gamma, \cot\alpha, \cot\alpha\cot\beta$ are in?

  1. A.P.

  2. G.P.

  3. H.P.

  4. A.G.P.
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A Correct answer
Explanation

Let the pole height be h and distances be x1, x2, x3 in A.P. Then cot(alpha) = x1/h, cot(beta) = x2/h, cot(gamma) = x3/h. Since x1, x2, x3 are in A.P., cot(alpha), cot(beta), cot(gamma) are in A.P. The sequence cot(beta)cot(gamma), cot(gamma)cot(alpha), cot(alpha)cot(beta) is a standard property related to A.P. terms.

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