Question

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

• A. less than 1
• B. equal to 1
• C. greater than 1
• D. any real number

Answer 1

The correct option is A

less than 1

Since a and b are unequal , $\frac{a^2+b^2}{2}>\sqrt{a^2{b}^2}$

(A.M. > G.M. for unequal numbers)

$\Rightarrow a^2+b^2>2ab$

Similarly $b^2+c^2$ > 2bc and $c^2+a^2>2ca$

Hence 2 $(a^2+b^2+c^2)>2(ab+bc+ca)$

$\Rightarrow ab+bc+ca<1$