Three vertical poles of heights `h_1, h_2 and h_3` at the vertices A, B and C of a `angleABC` subtend angles `alpha,beta `and `gamma` respectively at the cicumcentre of triangle. If `cotalpha, cotbeta` and `cotgamma` are in A.P. then `h_1,h_2,h_3,` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. none of these

Answer

A

Explanation

Let R be the circum-radius and O be the circumcentre of `triangleABC`.
In `triangleOAP_1`, we have
`tan alpha =(AP_1)/(OA) rArr tan alpha =h_1/R rArr h_1 =R cot alpha`
Similarly , we have
`h_2=R cot beta` and `h_3=R cot gamma`.

It is given that `cot alpha , cot beta, cot gamma` are in A.P.
`therefore 2 cot beta = cot alpha + cot gamma`
`rArr 2R cot beta = R cot alpha + R cot gamma`
`rArr 2h_2=h_1+h_3`
`rArr h_1,h_2,h_3` are in A.P.