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:

$3 M^2$

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