Question

If $b$ is the arithmetic mean between $a$ and $x$; $b$ is the geometric mean between ' $a$ ' and $y$; ' $b$ ' is the harmonic mean between a and $z,(a, b, x, y, z>0)$ then the value of $x y z$ is

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

Answer 1

Answer: (D)

$2 b = x + a $....(1)
$b ^2= ay $....(2)
$b =\frac{2 az }{ a + z } $....(3)
$x =2 b - a ; y =\frac{ b ^2}{ a } \text { and } \frac{2}{ b }=\frac{1}{ a }+\frac{1}{ z } \Rightarrow z =\frac{ ab }{2 a - b }$
$\therefore x y z=(2 b-a) \frac{b^2}{a} \cdot \frac{a b}{2 a-b}=\frac{b^3(2 b-a)}{2 a-b} $