# If a, b and c are distinct positive real numbers and a^2 + b^2 + c^2 = 1 , then ab + bc + ca is

Q: If a, b and c are distinct positive real numbers and a2 + b2 + c2 = 1 , then
ab + bc + ca is

(A) less than 1

(B) equal to 1

(C) greater than 1

(D) any real number

Ans: (A).

Sol: Since a and b are unequal, $\large \frac{a^2 + b^2}{2} > \sqrt{a^2 b^2}$ ( A.M. > G.M. for unequal numbers)

⇒ a2 + b2 > 2ab

Similarly , b2 + c2 > 2bc

and , c2 + a2 > 2ca.

Hence , 2(a2 + b2 + c2) > 2(ab + bc +ca)

⇒ ab + bc + ca < 1