Question

lf $a,b,c$ are non-zero real numbers and the sum of the squares of the roots of the equation $a{x}^2+bx+c=0$ is equal to the sum of the reciprocals of its roots, then $\frac{a}{b},\frac{b}{c},\frac{c}{a}$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. A.G.P

Answer 1

Answer: (C)

H.P.

Let $\alpha ,\beta$ be the roots of the equation $a{x}^2+bx+c=0$

$\alpha +\beta =-\frac{b}{a},\alpha \beta =\frac{c}{a}$

Given, ${\alpha }^2+{\beta }^2=\frac{1}{\alpha }+\frac{1}{\beta }$

$(\alpha +\beta {)}^2-2\alpha \beta =\frac{\alpha +\beta }{\alpha \beta }$

$\frac{b^2}{a^2}-\frac{2c}{a}=\frac{\frac{-b}{a}}{\frac{c}{a}}$

$\frac{b^2-2ac}{a^2}=\frac{-b}{c}$

$b^{2}c+a^{2}b=2a{c}^2$

$\Rightarrow \frac{b}{a}+\frac{a}{c}=\frac{2c}{b}$

$\Rightarrow \frac{b}{a},\frac{c}{b},\frac{a}{c}$ are in A.P.

$\therefore \frac{a}{b},\frac{b}{c},\frac{c}{a}$ are in H.P