If a - 1/a = b,\,b - 1/b = c and c - 1/c = a, then what is the value 1ab + 1bc + 1ca = ?

Option A. -3. Answer & Solution Answer: Option A Solution: & a - 1/a = b\,........ ( i ) & b - 1/b = c\,........ ( ii ) & c - 1/c = a\,........ ( iii ) & Add Equation ( i ),\, ( ii ) and ( iii ) & a + b + c - [ 1/a + 1/b + 1/c ] = a + b + c & 1/a + 1/b + 1/c = 0\,........ ( iv ) & a - 1/a = b…

If a - 1/a = b,\,b - 1/b = c and c - 1/c = a, then what is the value…

Answer: Option A

Solution:

a - \frac{1}{a} = b ........ (i) b - \frac{1}{b} = c ........ (ii) c - \frac{1}{c} = a ........ (iii) Add Equation (i), (ii) and (iii) a + b + c - [\frac{1}{a} + \frac{1}{b} + \frac{1}{c}] = a + b + c \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0 ........ (iv) \Rightarrow a - \frac{1}{a} = b Squaring both sides a^{2} + \frac{1}{a ^{2}} - 2 = b^{2} ........ (v) \Rightarrow b - \frac{1}{b} = c b^{2} + \frac{1}{b ^{2}} - 2 = c^{2} ........ (vi) \Rightarrow c + \frac{1}{c} = a c^{2} + \frac{1}{c ^{2}} - 2 = a^{2} ........ (vii) Add Equation (v), (vi) and (vii) a^{2} + b^{2} + c^{2} + \frac{1}{a ^{2}} + \frac{1}{b ^{2}} + \frac{1}{c ^{2}} - 2 - 2 - 2 = a^{2} + b^{2} + c^{2} \frac{1}{a ^{2}} + \frac{1}{b ^{2}} + \frac{1}{c ^{2}} = 6 ........ (viii) \Rightarrow (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) = \frac{1}{a ^{2}} + \frac{1}{b ^{2}} + \frac{1}{c ^{2}} + \frac{2}{ab} + \frac{2}{b c} + \frac{2}{c a} 0^{2} = 6 + 2 [\frac{1}{ab} + \frac{1}{b c} + \frac{1}{c a}] \frac{1}{ab} + \frac{1}{b c} + \frac{1}{c a} = \frac{- 6}{2} = - 3