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. $A{x}^2-(A^2+G^2)x+A{G}^2=0$
• B. $A{x}^2-(A^2+H^2)x+A{H}^2=0$
• C. $H{x}^2-(H^2+G^2)x+H{G}^2=0$
• D. $G{x}^2-(H^2+G^2)x+G{H}^2=0$

Answer 1

Answer: (A), (C)

The correct options are
A $A{x}^2-(A^2+G^2)x+A{G}^2=0$
C $H{x}^2-(H^2+G^2)x+H{G}^2=0$

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
$x^2-(A+H)x+AH=0$
Substituting $H=\frac{G^2}{A}$, we get
$x^2-(A+\frac{G^2}{A})x+G^2=0$

$\Rightarrow A{x}^2-(A^2+G^2)x+A{G}^2=0$

Hence, option A and similarly option C.