Question

If $a$ is the A.M. of $3$ numbers and $b$ is the A.M. of their squares, then the A.M. of their pair-wise products in terms of $a$ and $b$ is _____.

• A. $\frac{3a^2+b}{2}$
• B. $\frac{3a^2-b}{2}$
• C. $\frac{b-3a^2}{2}$
• D. $\frac{b^2-3a^2}{2}$

Answer 1

The correct option is B $\frac{3a^2-b}{2}$

Let the $3$ numbers be $x,y,z$

A.M. $=\frac{x+y+z}{3}=a$

$\Rightarrow 3a=x+y+z$

A.M. of their squares $=b$

$\Rightarrow b=\frac{x^2+y^2+z^2}{3}$

$\therefore 3b=x^2+y^2+z^2$

From formula,

$(x+y+z{)}^2=x^2+y^2+z^2+2xy+2yz+2zx$

$\therefore (3a{)}^2-3b=2(xy+yz+zx)$

$\therefore xy+yz+zx=\frac{9a^2-3b}{2}$

A.M. of pair-wise products $=\frac{xy+yz+zx}{3}=\frac{3a^2-b}{2}$