SATHEE: Sequences and Series 3 Question 2

Sequences and Series 3 Question 2

2. If three distinct numbers $a, b$ and $c$ are in GP and the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root, then which one of the following statements is correct?

(2019 Main, 8 April II)

(a) $d, e$ and $f$ are in GP

(b) $\frac{d}{a}, \frac{e}{b}$ and $\frac{f}{c}$ are in AP

(d) $\frac{d}{a}, \frac{e}{b}$ and $\frac{f}{c}$ are in GP

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Answer:

Correct Answer: 2. (b)

Solution:

(b) Given, three distinct numbers $a, b$ and $c$ are in GP. $∴ b^{2} = a c$ and the given quadratic equations

$\begin{aligned} a x^{2} + 2 b x + c = 0 \\ d x^{2} + 2 e x + f = 0 \end{aligned}$

For quadratic Eq. (ii), the discriminant $D = (2 b)^{2} - 4 a c$

$= 4 (b^{2} - a c) = 0$

$\Rightarrow$ Quadratic Eq. (ii) have equal roots, and it is equal to $x = - \frac{b}{a}$ , and it is given that quadratic Eqs. (ii) and (iii) have a common root, so

$\begin{array}{rcr} d - {\frac{b}{a}}^{2} + 2 e - \frac{b}{a} + f = 0 \\ \Rightarrow & d b^{2} - 2 e b a + a^{2} f = 0 & [∵ b^{2} = a c] \\ \Rightarrow & d (a c) - 2 e a b + a^{2} f = 0 & [∵ a \neq 0] \\ \Rightarrow & d c - 2 e b + a f = 0 \\ \Rightarrow & 2 e b = d c + a f \\ \Rightarrow & 2 \frac{e}{b} = \frac{d c}{b^{2}} + \frac{a f}{b^{2}} \\ 2 \frac{e}{b} = \frac{d}{a} + \frac{f}{c} & [∵ b^{2} = a c] \end{array}$

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

Alternate Solution

Given, three distinct numbers $a, b$ and $c$ are in GP. Let $a = a, b = a r, c = a r^{2}$ are in GP, which satisfies $a x^{2} + 2 b x + c = 0$

$\begin{array}{ll} ∴ & a x^{2} + 2 (a r) x + a r^{2} = 0 \\ \Rightarrow & x^{2} + 2 r x + r^{2} = 0 \\ \Rightarrow & (x + r)^{2} = 0 \Rightarrow x = - r . \end{array}$

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

So, $x = - r$ satisfies $d x^{2} + 2 e x + f = 0$

$∴ d (- r)^{2} + 2 e (- r) + f = 0$

$\Rightarrow d r^{2} - 2 e r + f = 0$

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

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

$\Rightarrow \frac{d}{a} + \frac{f}{c} = \frac{2 e}{b}$

$[∵ c \neq 0]$

Sequences and Series 3 Question 2

2. If three distinct numbers $a, b$ and $c$ are in GP and the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root, then which one of the following statements is correct?

Answer:

Alternate Solution

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Sequences and Series 3 Question 2

2. If three distinct numbers a,b and c are in GP and the equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common root, then which one of the following statements is correct?

Answer:

Alternate Solution

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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2. If three distinct numbers $a, b$ and $c$ are in GP and the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root, then which one of the following statements is correct?