Question

if A be the A.M and h be the H.M between two numbers a and b then show that a-A/a-H*b-A/b-H=A/H
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Answer 1

we have to show that,

(a - A)/(a - H) × (b - A)/(b - H) = A/H

arithmetic mean of a and b, A = (a + b)/2

harmonic mean of a and b, H = 2ab/(a + b)

LHS = {a - (a + b)/2}/{a - 2ab/(a + b)} × {b - (a + b)/2}/{b - 2ab/(a + b)}

= {2a - a - b}(a + b)/2(a² + ab - 2ab) × {2b - a - b}(a + b)/2(ba + b² - 2ab)

= (a - b)(a + b)/2(a² - ab) × (b - a)(a + b)/2(b² - ab)

= (a + b)/2a × (a + b)/2b

= {(a + b)/2}/{2ab/(a + b)}

= A/H = RHS

hence proved

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