If a + b + c = 1 and ab + bc + ca = ⅓ then a : b : c is

A 1 : 2 : 2

B 2 : 1 : 2

C 1 : 1 : 1

D 1 : 2 : 1

E 1 : 3 : 1

Correct Answer: Option C

Given that, a + b + c = 1 and ab + bc + ca = ⅓

Then, \(\frac{{ab\; + \;bc\; + \;ca}}{{a + b + c}} = \frac{1}{3}\)

⇒ 3ab + 3bc + 3ca = a + b + c

⇒ a(3b – 1) + b(3c – 1) + c(3a – 1) = 0

Since a, b, c are non-zero terms,

⇒ (3b – 1), (3c - 1) and (3a - 1) must be zero

⇒ a = b = c = ⅓

⇒ a : b : c = ⅓ : ⅓ : ⅓

Or, a : b : c = 1 : 1 : 1