If a, b, c be positive real number and the value of θ = tan-1 √a (a + b + c) / bc + tan-1 √b (a + b + c) / ca + tan-1 √c (a + b + c) / ab, then tan θ =

1) 0

2) 1

3) [a + b + c] / abc

4) None of these

Solution: (1) 0

θ = tan-1 √a (a + b + c) / bc + tan-1 √b (a + b + c) / ca + tan-1 √c (a + b + c) / ab

Let s2 = [a + b + c] / abc

θ = tan-1 √a2s2 + tan-1 √b2s2 + tan-1 √c2s2

= tan-1 as + tan-1 bs + tan-1 cs

= tan-1 [as + bs + cs – abcs3] / [1 – abs2 – acs2 – bcs2]

tan θ = s [(a + b + c) – (a + b + c)] / [1 – s2 (ab + bc + ca)] = 0

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