Question

$Iftheroots\alpha ,\beta ,\gamma oftheequation{x}^3-{3ax}^2+3bx+c=0areinH.Pthen$

• A. $\beta =\frac{1}{\alpha }$
• B. $\beta =b$
• C. $\beta =\frac{-c}{b}$
• D. $\beta =\frac{b}{c}$

Answer 1

Answer: (C)

$\beta =\frac{-c}{b}$

$\alpha ,\beta$ and $\gamma$ are the roots of $x^3+3a{x}^3+3bx+c=0.$

$\alpha +\beta +\gamma =-3a$

$\alpha \beta +\beta \gamma +\alpha \gamma =3b$

$\alpha \beta \gamma =-c$

$\frac{1}{\alpha },\frac{1}{\beta },\frac{1}{\gamma }$ are in $A.P$, If $\alpha ,\beta ,\gamma$ are in $H.P.$

$\frac{1}{\alpha }+\frac{1}{\gamma }=\frac{2}{\beta }$

$\frac{\alpha +\gamma }{\alpha \gamma }=\frac{2}{\beta }$

$\alpha \beta +\beta \gamma =2\alpha \gamma$

$\alpha \beta +\beta \gamma +\alpha \gamma =3\alpha \gamma =\frac{3\alpha \beta \gamma }{\beta }$

$3b=-\frac{3c}{\beta }$

$\beta =-\frac{c}{b}$