In a triangle `ABC, if cotA :cotB :cotC = 30: 19 : 6` then the sides `a, b, c` are a) in A.P. b) in G.P. c) in H.P. d) None of these

Options

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

Answer

A

Explanation

`cot A = (cos A)/(sin A) = ((b^(2) + c^(2) -a^(2))/(2bc))/((a)/(2R)) = ((b^(2) + c^(2) -a^(2))/(abc)) R`
`:. cot A : cot B : cot C`
`= b^(2) + c^(2) -a^(2) : c^(2) + a^(2) - b^(2) : a^(2) + b^(2) -c^(2)`
Clearly if `cot A : cot B : cot C = l : m : n`
then `a: b : c = sqrt(m + n) : sqrt(n +l) : sqrt(l + m)`
`= sqrt(19 + 6) : sqrt(6 + 30) : sqrt(30 + 19)`
`= 5 : 6 : 7`
Therefore, a, b, c are in A.P.