SATHEE: Sequences and Series 3 Question 12

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) $A P$

(b) $G P$

(d) None of these

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Answer:

Correct Answer: 12. (a)

Solution:

Since, $a, b, c$ are in GP.

$\Rightarrow b^{2} = a c$

Given, $a x^{2} + 2 b x + c = 0$

$\Rightarrow a x^{2} + 2 \sqrt{a c} x + c = 0$

$\Rightarrow (\sqrt{a} x + \sqrt{c})^{2} = 0 \Rightarrow x = - \sqrt{\frac{c}{a}}$

Since, $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have common root.

$∴ x = - \sqrt{c / a}$ must satisfy.

$d x^{2} + 2 e x + f = 0$

$\Rightarrow d \cdot \frac{c}{a} - 2 e \sqrt{\frac{c}{a}} + f = 0 \Rightarrow \frac{d}{a} - \frac{2 e}{\sqrt{a c}} + \frac{f}{c} = 0$

$\Rightarrow \frac{2 e}{b} = \frac{d}{a} + \frac{f}{c} [∵ b^{2} = a c]$

Hence, $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in an AP.

Sequences and Series 3 Question 12

12.

Answer:

Engineering Preparation

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Mentorship

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Sequences and Series 3 Question 12

12.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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