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

Solution

## The correct options are A $$Ax^{2}-({A}^{2}+G^{2})x+{A}G^{2}=0$$ C $$Hx^{2}-(H^{2}+G^{2})x+HG^{2}=0$$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 $${ G }^{ 2 }=AH$$Now equation whose roots are $$A$$ and $$H$$ is$${ x }^{ 2 }-\left( A+H \right) x+AH=0$$Substituting $$H=\cfrac { { G }^{ 2 } }{ A }$$, we get$${ x }^{ 2 }-\left( A+\cfrac { { G }^{ 2 } }{ A } \right) x+ G^2 =0$$$$\Rightarrow A{ x }^{ 2 }-\left( { A }^{ 2 }+{ G }^{ 2 } \right) x+A{ G }^{ 2 }=0$$Hence, option A and similarly option C.Maths

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