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Question

If the A.M. and G.M. between two numbers are in ratio of m:n, then the ratio of numbers can be 
  1. m+m2n2mm2n2
  2. m+m2n2nm2n2 
  3. m+m2+n2mm2+n2
  4. n+m2n2mm2n2


Solution

The correct option is A m+m2n2mm2n2
Let the two numbers be a and b, so
A=a+b2,  G=aba+b=2A,  ab=G2(1)

The equation whose roots are a and b is
x2(a+b)x+ab=0x22Ax+G2=0x=2A±4A24G22x=A±A2G2

So, the two numbers are a=A+A2G2 and
b=AA2G2
It is given that
A:G=m:nA=km,G=kn
Substituting the values of A and G, we get
ab=km+k2m2k2n2kmk2m2k2n2ab=m+m2n2mm2n2

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