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Question

Let A,G and H be the AM, GM and HM of two positive numbers a and b. The quadratic equation whose roots are A and H is

A
Ax2(A2+G2)x+AG2=0
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B
Ax2(A2+H2)x+AH2=0
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C
Hx2(H2+G2)x+HG2=0
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D
Gx2(H2+G2)x+GH2=0
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Solution

The correct options are
A Ax2(A2+G2)x+AG2=0
C Hx2(H2+G2)x+HG2=0
As A,G,H are A.M, G.M and H.M between a and b.
Then A,G,H is in G.P such that G2=AH
Now equation whose roots are A and H is
x2(A+H)x+AH=0
Substituting H=G2A, we get
x2(A+G2A)x+G2=0
Ax2(A2+G2)x+AG2=0

Hence, option A and similarly option C.

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