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Relation between AM,GM,HM for 2 Numbers

If b is the...

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 ?

\frac{(2 b - a) b^{3}}{b - 2 a}

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_{b} 3

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\frac{b^{3} (2 a - b)}{2 b - a}

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\frac{b^{3} (2 a - a)}{2 a - b}

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Solution

The correct option is B $\frac{(2 b - a) b^{3}}{b - 2 a}$
wkt, $b = \frac{a + x}{2}$

$x = 2 b - a$

$b = \sqrt{a y}$

$y = \frac{b^{2}}{a}$

$b = \frac{2 a z}{a + z}$

$a b + z b = 2 a z$

$z = \frac{a b}{b - 2 a}$

$x y z = (2 b - a) * (\frac{a b}{b - 2 a}) * \frac{b^{2}}{a}$

$= \frac{(2 b - a) b^{3}}{b - 2 a}$

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