a,g,h are arithmetic mean, geometric mean and harmonic mean between two positive numbers x and y respectively. Then identify the correct statement among the following

h is the harmonic mean between a and g

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No such relation exists between a,g and h

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q is the geometric mean between a and h

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A is the arithmetic mean between g and h

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Solution

The correct option is D A is the arithmetic mean between g and h
$a = \frac{x + y}{2}, g = \sqrt{x y}$ and $h = \frac{2 x y}{x + y}$
$g^{2} = (\sqrt{x y})^{2} = x y$ ...(i), $a h = \frac{x + y}{2} . \frac{2 x y}{x + y} = x y . . . . (i i)$
From (i) and (ii), we get $g^{2} = a h$ or $g = \sqrt{a h}$ , therefore g is the G.M. between a and h.

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a,g,h are arithmetic mean, geometric mean and harmonic mean between two positive numbers x and y respectively. Then identify the correct statement among the following

The correct option is D A is the arithmetic mean between g and h a=x+y2,g=√xy and h=2xyx+y g2=(√xy)2=xy ...(i), ah=x+y2.2xyx+y=xy....(ii) From (i) and (ii), we get g2=ah or g=√ah, therefore g is the G.M. between a and h.