Question

If the mean of a,b,c is M and ab + bc + ca = 0, then mean of a^2,b^2,c^2 is

Answer 1

Mean of a,b,c is M ;
Hence ;
(a+b+c)/3 = M
a+b+c = 3M ------------------ (i)
=======================
& ab+bc+ca = 0
____________________
Now, Mean of a^2, b^2,c^2 ;

= (a^2+b^2+c^2)/3
= [(a+b+c)^2 -2ab-2bc-2ca]/3
= [(3M)^2 -2(ab+bc+ca)]/3
={ 9M^2 - 2(0) } ÷3
= (9M^2)/3

= 3M^2
_____________________

Answer 2

Mean of a,b,c = M
a+b+c/3 = M
a+b+c = 3M

Squaring on both sides.
(a + b + c)² =(3M)²
a²+b²+c² + 2(ab+bc+ca) = 3M²
a²+b²+c²+2(0) = 3M². [ab + bc + ca = 0]
a²+b²+c² = 3M²

Mean of a²+b²+c² = a²+b²+c²/3
=>3M²/3. [ a²+b²+c² = 3M²]
=>M²