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If the mean of a, b, c, is M and ab + bc + ca = 0, then the mean of a2, b2, c2 is -
Answer & Solution
Correct Answer:
Option
B
We have :
$$\frac{a + b + c}{3}$$ = M
or (a + b + c) = 3M
Now, (a + b + c)2 = (3M)2 = 9M2
⇔ a2 + b2 + c2 + 2 (ab + bc + ca) = 9M2
⇔ a2 + b2 + c2 = 9M2
[$$\because $$ (ab + bc + ca) = 0]
∴ Required mean
$$\eqalign{ & = \left( {\frac{{{a^2} + {b^2} + {c^2}}}{3}} \right) \cr & = \frac{{9{M^2}}}{3} \cr & = 3{M^2} \cr} $$
$$\frac{a + b + c}{3}$$ = M
or (a + b + c) = 3M
Now, (a + b + c)2 = (3M)2 = 9M2
⇔ a2 + b2 + c2 + 2 (ab + bc + ca) = 9M2
⇔ a2 + b2 + c2 = 9M2
[$$\because $$ (ab + bc + ca) = 0]
∴ Required mean
$$\eqalign{ & = \left( {\frac{{{a^2} + {b^2} + {c^2}}}{3}} \right) \cr & = \frac{{9{M^2}}}{3} \cr & = 3{M^2} \cr} $$
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