If the mean of a, b, c is M and $ab+bc+ca=0$, then the mean of $a^2, ~b^2$ and $c^2$ will be:
Answer & explanation
Correct answer: option 2
We have,
Average of a , b ,c = M
& ab + bc + ca = 0
We know that,
Average of values = ( sum of values ) / Number of values
Average of a , b , c = M
i.e. (a + b + c) / 3 = M
⇒ a + b + c = 3M
On squaring both side :
⇒ (a + b + c)2 = (3M)2
⇒ a2 + b2 + c2 + 2 (ab + bc + ca) = 9M2 ................ Eq (1)
Now putting the value of ( ab + bc + ca ) = 0 in Eq (1)
⇒ a2 + b2 + c2 = 9M2
Now average of a2 + b2 + c2 : = 9M2 / 3 = 3M2