
If \[\frac{{a + bx}}{{a - bx}} = \frac{{b + cx}}{{b - cx}} = \frac{{c + dx}}{{c - dx}}(x \ne 0)\], then \[a,b,c,d\] are in
A. A.P.
B. G.P.
C. H.P.
D. None of these
Answer
162.9k+ views
Hint
The proportions theorem of "componendo and dividendo" allows fast calculations and minimizes the quantity of expansions needed. It's notably useful for determination three-quarter equations or rational equations in mathematical Olympiads, particularly once fractions area unit gift. A proportion is Associate in Nursing equation that sets 2 ratios at constant price.
If each part of the set has Associate in Nursing inverse for a selected operation, the set has the inverse attribute. Associate in Nursing part's inverse may be a completely different element within the set that, once coupled on the correct or left by Associate in Nursing operation, invariably yields the operator. The phrase "when we tend to multiply any variety by zero, the ensuing product is usually a zero" is employed to outline the zero property of multiplication.
Formula used:
\[a,b,c\] in G.P
\[ {b^2} = ac\]
Complete step-by-step solution
The given equation is
\[\frac{{a + bx}}{{a - bx}} = \frac{{b + cx}}{{b - cx}}\]
Cross multiply the equation
\[(a + bx)(b - cx) = (b + cx)(a - bx)\]
\[ = > ab + {b^2}x - acx - bc{x^2} = ab + acx - {b^2}x - bc{x^2}\]
It can be solved as
\[ = > 2{b^2}x = 2acx\]
\[ = > {b^2} = ac\] ---(1)
Another equation given is
\[\frac{{b + cx}}{{b - cx}} = \frac{{c + dx}}{{c - dx}}\]
Cross multiply this equation to get
\[(b + cx)(c - dx) = (c + dx)(b - cx)\]
\[ = > bc + {c^2}x - bdx - cd{x^2} = bc + bdx - {c^2}x - cd{x^2}\]
\[ = > 2{c^2}x = 2bdx\]
\[ = > {c^2} = bd\] ---(2)
\[a,b,c,d\] in G.P from the equations (1) and (2)
Therefore, the correct option is B.
Note
A mathematical sequence known as a geometric progression (GP) is one in which each following phrase is generated by multiplying each preceding term by a fixed integer, or "common ratio." This progression is sometimes referred to as a pattern-following geometric sequence of numbers. In an arithmetic progression (AP), the differences between every two consecutive terms are all the same, whereas in a geometric progression, the ratios of every two successive terms are all the same (GP).
The proportions theorem of "componendo and dividendo" allows fast calculations and minimizes the quantity of expansions needed. It's notably useful for determination three-quarter equations or rational equations in mathematical Olympiads, particularly once fractions area unit gift. A proportion is Associate in Nursing equation that sets 2 ratios at constant price.
If each part of the set has Associate in Nursing inverse for a selected operation, the set has the inverse attribute. Associate in Nursing part's inverse may be a completely different element within the set that, once coupled on the correct or left by Associate in Nursing operation, invariably yields the operator. The phrase "when we tend to multiply any variety by zero, the ensuing product is usually a zero" is employed to outline the zero property of multiplication.
Formula used:
\[a,b,c\] in G.P
\[ {b^2} = ac\]
Complete step-by-step solution
The given equation is
\[\frac{{a + bx}}{{a - bx}} = \frac{{b + cx}}{{b - cx}}\]
Cross multiply the equation
\[(a + bx)(b - cx) = (b + cx)(a - bx)\]
\[ = > ab + {b^2}x - acx - bc{x^2} = ab + acx - {b^2}x - bc{x^2}\]
It can be solved as
\[ = > 2{b^2}x = 2acx\]
\[ = > {b^2} = ac\] ---(1)
Another equation given is
\[\frac{{b + cx}}{{b - cx}} = \frac{{c + dx}}{{c - dx}}\]
Cross multiply this equation to get
\[(b + cx)(c - dx) = (c + dx)(b - cx)\]
\[ = > bc + {c^2}x - bdx - cd{x^2} = bc + bdx - {c^2}x - cd{x^2}\]
\[ = > 2{c^2}x = 2bdx\]
\[ = > {c^2} = bd\] ---(2)
\[a,b,c,d\] in G.P from the equations (1) and (2)
Therefore, the correct option is B.
Note
A mathematical sequence known as a geometric progression (GP) is one in which each following phrase is generated by multiplying each preceding term by a fixed integer, or "common ratio." This progression is sometimes referred to as a pattern-following geometric sequence of numbers. In an arithmetic progression (AP), the differences between every two consecutive terms are all the same, whereas in a geometric progression, the ratios of every two successive terms are all the same (GP).
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Main 2025 Session 2: Exam Date, Admit Card, Syllabus, & More

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

Trending doubts
Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Main Chemistry Question Paper with Answer Keys and Solutions

JEE Main Reservation Criteria 2025: SC, ST, EWS, and PwD Candidates

What is Normality in Chemistry?

Chemistry Electronic Configuration of D Block Elements: JEE Main 2025

Other Pages
NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations

NCERT Solutions for Class 11 Maths Chapter 8 Sequences and Series

Total MBBS Seats in India 2025: Government College Seat Matrix

NEET Total Marks 2025: Important Information and Key Updates

Neet Cut Off 2025 for MBBS in Tamilnadu: AIQ & State Quota Analysis

Karnataka NEET Cut off 2025 - Category Wise Cut Off Marks
