Doubt by Pari
Solution :
Mean of a,b and c = M (Given)
ab+bc+ca=0 (Given)
ab+bc+ca=0 (Given)
Mean of a²+b²+c² = ?
We know,
Mean (x̄)
Mean (x̄)
= [Sum of Observations] / [Total Number of observations]
ATQ
M = (a+b+c)/3
3M = a+b+c
3M = a+b+c
S.B.S
(3M)²=(a+b+c)²
9M²=a²+b²+c²+2ab+2bc+2ca
[∵ (a+b+c)²=a²+b²+c²+2ab+2bc+2ca]
9M²=a²+b²+c²+2ab+2bc+2ca
[∵ (a+b+c)²=a²+b²+c²+2ab+2bc+2ca]
9M²=a²+b²+c²+2(ab+bc+ca)
9M²=a²+b²+c²+2(0) [∵ab+bc+ca=0]
9M²=a²+b²+c²+0
9M²=a²+b²+c²+2(0) [∵ab+bc+ca=0]
9M²=a²+b²+c²+0
9M²=a²+b²+c² — (1)
Now,
Mean of a²+b²+c²
Mean of a²+b²+c²
= (a²+b²+c²)/3
= 9M²/3 [ From equation (1) ]
= 9M²/3 [ From equation (1) ]
= 3M²
Hence, If the mean of a, b, c is M and ab+bc+ca=0, then the mean of a²+b²+c² is 3M².