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 $a/c,b/a,c/b$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

Answer: (C)

H.P.

Answer 2

As given, if α, β be the roots of the quadratic equation, then

⇒ $\alpha + \beta = \frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}} = \frac{(\alpha + \beta)^{2} - 2\alpha\beta}{\alpha^{2}\beta^{2}}$

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

⇒ $\frac{2a}{c} = \frac{b^{2}}{c^{2}} + \frac{b}{a} = \frac{ab^{2} + bc^{2}}{ac^{2}}$ ⇒ $2a^{2}c = ab^{2} + bc^{2}$

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

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