Question
If the sum of the roots of the quadratic equation $$a{x^2} + bx + c = 0$$ is equal to the sum of the squares of their reciprocals, then $$\frac{a}{c},\frac{b}{a}\,\,{\text{and}}\,\frac{c}{b}$$ are in
A.
Arithmetic - Geometric Progression
B.
Arithmetic Progression
C.
Geometric Progression
D.
Harmonic Progression.
Answer :
Harmonic Progression.
Solution :
$$a{x^2} + bx + c = 0,\,\,\alpha + \beta = \frac{{ - b}}{a},\alpha \beta = \frac{c}{a}$$
As for given condition, $$\alpha + \beta = \frac{1}{{{\alpha ^2}}} + \frac{1}{{{\beta ^2}}}$$
$$\eqalign{
& \alpha + \beta = \frac{{{\alpha ^2} + {\beta ^2}}}{{{\alpha ^2}{\beta ^2}}} - \frac{b}{a} = \frac{{\frac{{{b^2}}}{{{a^2}}} - \frac{{2c}}{a}}}{{\frac{{{c^2}}}{{{a^2}}}}} \cr
& {\text{On simplification }}2{a^2}c = a{b^2} + b{c^2} \cr
& \Rightarrow \,\,\frac{{2a}}{b} = \frac{c}{a} + \frac{b}{c} \cr
& \Rightarrow \,\,\frac{c}{a},\frac{a}{b},\frac{b}{c}\,\,{\text{are in A}}{\text{.P}}{\text{.}} \cr
& \therefore \,\,\frac{a}{c},\frac{b}{a}\& \frac{c}{b}\,\,{\text{are in H}}{\text{.P}}{\text{.}}\,\, \cr} $$