If the mean of a, b, c is M and ab + bc + ca = 0, then the mean of a2+b2+c23 is

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Question
If the mean of a, b, c is M and ab + bc + ca = 0, then the mean of a2+b2+c23 is :
Options
A. M
B. 3 M^2
C. 2 (M/3)
D. M^2

Correct Answer

3 M^2

Explanation

We have : ( a + b + c) / 3 = M or (a + b + c) = 3M.

Now. ${(a + b + c)}^{2} = {(3 M)}^{2} = 9 M^{2}$ .

$< = > a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a) = 9 M^{2}$

$a^{2} + b^{2} + c^{2} = 9 M^{2}, (a b + b c + c a = 0)$

Required mean = $(\frac{a^{2} + b^{2} + c^{2}}{3}) = \frac{9 M^{2}}{3} = 3 M^{2}$ .

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