If a – (1/a) = b, b – (1/b) = c and c – (1/c) = a, then what is the value of (1/ab) + (1/bc) + (1/ca)?
Show answer & explanation
Correct answer: A
Explanation
The correct answer is option 1 i.e. -3
a – (1/a) = b, b – (1/b) = c and c – (1/c) = a,
(a2 - 1)/a = b
a2 - 1 = ab .....(1)
b2 - 1 = bc .....(2)
c2 - 1 = ac .....(3)
adding (1) + (2) + (3)
a2 + b2 + c2 - 3 = ab + bc + ac .....(4)
= (1/ab) + (1/bc) + (1/ca)
= (a - b)(b - c) + (b - c)(c - a) + (c - a)(a - b)
= ab - ac - b2 + bc + bc - ab - c2 + ac + ac - bc - a2 + ab
= -a2 - b2 - c2 + ab + bc + ca
= -(a2 + b2 + c2 - ab - bc - ca)
= -3