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If A ,G, H  be he arithematic geometric and harmonic mean of two numbers a and b then quadratic equation whose roots are A And H May be
Asked by harshitj291 | 10 Mar, 2019, 11:33: AM
answered-by-expert Expert Answer
A = (a+b)/2   ;  H = 2ab/(a+b)  ;  begin mathsize 12px style G space equals space square root of a space b end root end style
if roots are A and H,  then the quadtratic equation is , x2 - (A+H)x + (A×H) = 0
 
begin mathsize 12px style A plus H space equals space fraction numerator a plus b over denominator 2 end fraction plus fraction numerator 2 a space b over denominator open parentheses a plus b close parentheses end fraction space equals space fraction numerator open parentheses a plus b close parentheses squared plus 4 a space b over denominator 2 open parentheses a plus b close parentheses end fraction space equals space fraction numerator 4 A squared plus 4 G squared over denominator 4 A end fraction space equals space fraction numerator A squared plus G squared over denominator A end fraction
A cross times H space equals space fraction numerator a plus b over denominator 2 end fraction cross times fraction numerator 2 space a space b over denominator a plus b end fraction space equals space a space b space equals space G squared end style
 
Hence the quadratic eqn.  is  [  A x2 - ( A2 + G2 ) x + (A G2 ) ] = 0
Answered by Thiyagarajan K | 10 Mar, 2019, 03:25: PM

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