Sequences and Series 3 Question 12
12. If $a, b, c$ are in GP, then the equations $a x^{2}+2 b x+c=0$ and $d x^{2}+2 e x+f=0$ have a common root, if $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in
(1985, 2M)
(a) $AP$
(b) GP
(c) HP
(d) None of these
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Answer:
Correct Answer: 12. $(a=5)(b=8)(c=12)$
Solution:
- Since, $a, b, c$ are in GP.
$\Rightarrow \quad b^{2}=a c$
Given, $\quad a x^{2}+2 b x+c=0$
$\Rightarrow \quad a x^{2}+2 \sqrt{a c} x+c=0$
$\Rightarrow \quad(\sqrt{a} x+\sqrt{c})^{2}=0 \Rightarrow x=-\sqrt{\frac{c}{a}}$
Since, $a x^{2}+2 b x+c=0$ and $d x^{2}+2 e x+f=0$ have common root.
$\therefore \quad x=-\sqrt{c / a}$ must satisfy.
$\begin{aligned} & d x^{2}+2 e x+f & =0 \ \Rightarrow & d \cdot \frac{c}{a}-2 e \sqrt{\frac{c}{a}} & +f=0 \Rightarrow \frac{d}{a}-\frac{2 e}{\sqrt{a c}}+\frac{f}{c}=0 \ \Rightarrow & \frac{2 e}{b} & =\frac{d}{a}+\frac{f}{c} \quad\left[\because b^{2}=a c\right]\end{aligned}$
Hence, $\quad \frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in an AP.
