Question

If $A,G$ and $H$ are respectively the AM,GM and HM of three positive numbers $a,b$ and $c$; find the equation whose roots are $a,b$ and $c$

• A. $G{x}^3-3AG{x}^2+3A^{3}x-A^3{G}^3=0$
• B. $H{x}^3-3AH{x}^2+3G^{3}x-H{G}^3=0$
• C. $A{x}^3-3AG{x}^2+3G^{2}H-H^3$
• D. $A{x}^3+HG{x}^2-GHx+G=0$

Answer 1

The correct option is B $H{x}^3-3AH{x}^2+3G^{3}x-H{G}^3=0$

$A=\frac{(a+b+c)}{3}G={\left(abc\right)}^{1/3}H=\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}=\frac{3abc}{ab+bc+ac}(a+b+c)=3Aab+bc+ac=\left(\frac{3G^3}{H}\right)abc=G^3$

Equation having rooots $a,b,c$ are

$x^3-(a+b+c)x^2+(ab+bc+ac)x-abc=0x^3-3A{x}^2+\frac{3G^3}{H}x-g^3=0H{x}^3-3AH{x}^2+3G^{3}x-G^{3}H=0$