Geometric progression

If three distinct numbers a, b and c are in GP and the equatio

Question

Hard

Solving time: 5 mins

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?

d, e

and

f

are in

GP

\frac{d}{a}, \frac{e}{b}

and

\frac{f}{c}

are in

AP

d, e

and

f

are in

AP

\frac{d}{a}, \frac{e}{b}

and

\frac{f}{c}

are in GP

Views: 5,632 students

Updated on: Jul 5, 2023

Book:

Arihant JEE Main Chapterwise Solutions Mathematics (2019-2002) (Arihant)

Exam:

JEE Mains 2019

Text solutionVerified

Exp. (b)
Given, three distinct numbers

a, b

and

c

are in GP.

∴ b^{2} = a c

....(i)
and the given quadratic equations

a x^{2} + 2 b x + c = 0

...(ii)

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

....(iii)
For quadratic Eq. (ii),
the discriminant

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

= 4 (b^{2} - a c) = 0 [from Eq. (i)]

\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

d (- \frac{b}{a})^{2} + 2 e (- \frac{b}{a}) + f = 0

\Rightarrow d b^{2} - 2 e ba + a^{2} f = 0

\Rightarrow d (a c) - 2 e ab + a^{2} f = 0 [∵ b^{2} = a c]

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

[∵ a \neq = 0]

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

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

∴ 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

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 er + 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]

Was this solution helpful?

Report

Video solutions (19)

Learn from their 1-to-1 discussion with Filo tutors.

3 mins

Uploaded on: 10/23/2022

Ask your question, on a video call with tutor

Connect instantly with this tutor

Connect now

Taught by

Somesh Jain

Total classes on Filo by this tutor - 14,239

Connect instantly with this tutor

Connect now

Notes from this class (4 pages)

Download

Was this solution helpful?

Report

5 mins

Uploaded on: 4/27/2023

Ask your question, on a video call with tutor

Connect instantly with this tutor

Connect now

Taught by

Anuj Pratap

Total classes on Filo by this tutor - 2,989

Connect instantly with this tutor

Connect now

Notes from this class (4 pages)

Download

Was this solution helpful?

Report

2 mins

Uploaded on: 5/1/2023

Ask your question, on a video call with tutor

Connect instantly with this tutor

Connect now

Taught by

Pramod Singh

Total classes on Filo by this tutor - 16,104

Teaches : Mathematics, Algebra, Calculus

Connect instantly with this tutor

Connect now

Notes from this class (1 pages)

Download

Was this solution helpful?

113

Report

See more 1-on-1 discussions (16)

Found 4 tutors discussing this question

Lucas Discussed

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?

10 mins ago

Discuss this question LIVE

10 mins ago

One destination for complete JEE/NEET preparation

One destination to cover all your homework and assignment needs

Learn Practice Revision Succeed

Instant 1:1 help, 24x7

60, 000+ Expert tutors

Textbook solutions

Big idea maths, McGraw-Hill Education etc

Essay review

Get expert feedback on your essay

Schedule classes

High dosage tutoring from Dedicated 3 experts

Trusted by 4 million+ students

Stuck on the question or explanation?

Connect with our Mathematics tutors online and get step by step solution of this question.

231 students are taking LIVE classes

Question Text	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?
Updated On	Jul 5, 2023
Topic	Sequences and Series
Subject	Mathematics
Class	Class 11
Answer Type	Text solution:1 Video solution: 19
Upvotes	1750
Avg. Video Duration	7 min

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?

Text solutionVerified

Video solutions (19)

Questions from JEE Mains 2019 - PYQs

Practice questions from Arihant JEE Main Chapterwise Solutions Mathematics (2019-2002) (Arihant)

Practice questions from Sequences and Series in the same exam

Practice more questions from Sequences and Series

Practice questions on similar concepts asked by Filo students

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?

Text solutionVerified

Video solutions (19)

Questions from JEE Mains 2019 - PYQs

Practice questions from Arihant JEE Main Chapterwise Solutions Mathematics (2019-2002) (Arihant)

Practice questions from Sequences and Series in the same exam

Practice more questions from Sequences and Series

Practice questions on similar concepts asked by Filo students

Related books for questions on Sequences and Series

Previous year question papers

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?