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Question

If m is a root of the given equation (1ab)x2(a2+b2)x(1+ab)=0 and m harmonic means are inserted between a and b, then the difference between the last and the first of the means equals

A
b - a
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B
ab(b - a)
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C
a(b - a)
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D
ab(a - b)
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Solution

The correct option is B ab(b - a)
By the given condition
(1ab)m2(a2+b2)m(1+ab)=0
m(a2+b2)+(m2+1)ab=m21 ......(i)
Now H1 First H.M. between a and b

= (m+1)aba+mb and Hm=(m+1)abb+ma
HmH1=(m+1)ab[1b+ma1a+mb]
=(m+1)ab[(m1)(ba)](b+ma)(a+mb)=(m21)ab(ba)M(a2+b2)+(m2+1)ab
=(m21)ab(ba)m21 [by(i)]
= ab(b - a).


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