Question

If the sum of the roots of the quadratic equation $ax^2 + bx + c = 0$ is equal to the sum of the squares of their reciprocals, then $\frac{a}{c}, \frac{b}{a}$ and $\frac{c}{b}$ are in

• A. arithmetic progression
• B. geometric progression
• C. harmonic progression
• D. arithmetic−geometric−progression

Answer 1

Answer: (C)

$α + β =\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}
α + β =\frac{\alpha^{2}+\beta^{2}-2\alpha\beta}{\left(α + β \right)}$
$\left(-\frac{b}{a}\right)=\frac{b^{2}-2ac}{c^{2}}$
$\Rightarrow 2a^{2}c=b\left(a^{2}+bc\right)$
$\Rightarrow \frac{a}{c}, \frac{b}{a}, \frac{c}{b}$ are in $H.P.$