Question 13 Marks
If the mean of $a, b, c$ is $M$ and $a b+b c+c a=0$, then find the mean of $a^2, b^2, c^2$.
Answer
View full question & answer→We have : $\left(\frac{a+b+c}{3}\right)=M$
$
\begin{aligned}
or\ a+b+c =3 M \\
Now, (a+b+c)^2 & =9 M^2
\end{aligned}
$
$\begin{array}{l}\Rightarrow a^2+b^2+c^2+2(a b+b c+a c)=9 M^2 \\ \Rightarrow \quad a^2+b^2+c^2=9 M^2 \\ \quad[\because \text { Given, } a b+b c+c a=0]\end{array}$
$\therefore$ Required mean $=\left(\frac{a^2+b^2+c^2}{3}\right)=\frac{9 M^2}{3}=3 M^2$.
$
\begin{aligned}
or\ a+b+c =3 M \\
Now, (a+b+c)^2 & =9 M^2
\end{aligned}
$
$\begin{array}{l}\Rightarrow a^2+b^2+c^2+2(a b+b c+a c)=9 M^2 \\ \Rightarrow \quad a^2+b^2+c^2=9 M^2 \\ \quad[\because \text { Given, } a b+b c+c a=0]\end{array}$
$\therefore$ Required mean $=\left(\frac{a^2+b^2+c^2}{3}\right)=\frac{9 M^2}{3}=3 M^2$.