Q85 / 225 · JEE Main 2003 · Mathematics

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

  • A
    Arithmetic - Geometric Progression
  • B
    Arithmetic Progression
  • C
    Geometric Progression
  • Correct answer
    D
    Harmonic Progression

Solution

$a x^2+b x+c=0, \alpha+\beta=\frac{-b}{a}, \alpha \beta=\frac{c}{a}$ As for given condition, $\alpha+\beta=\frac{1}{\alpha^2}+\frac{1}{\beta^2}$ $\alpha+\beta=-\frac{\alpha^2+\beta^2}{\alpha^2 \beta^2}-\frac{b}{a}=\frac{\frac{b^2}{a^2}-\frac{2 c}{a}}{\frac{c^2}{a^2}}$ On simplification $2 a^2 c=a b^2+b c^2$ $\Rightarrow \frac{2 \mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{a}}+\frac{\mathrm{b}}{\mathrm{c}} \quad \therefore \frac{\mathrm{a}}{\mathrm{b}}, \frac{\mathrm{b}}{\mathrm{a}}, \& \frac{\mathrm{c}}{\mathrm{b}}$ are in H.P.

Hence, the correct answer is option: 'D'.

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