Class 11 Exam > Class 11 Questions > If a,b,c,d are in ap & a,c,d are in gp.So pro...
Start Learning for Free
If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad)?
Most Upvoted Answer
If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad...
Given Conditions
- Let a, b, c, d be in Arithmetic Progression (AP).
- This implies: 2b = a + d.
- Let a, c, d be in Geometric Progression (GP).
- This implies: c^2 = ad.
Expressing c in terms of a and d
- From the AP condition, we can express c as:
- c = (a + d)/2.
Substituting c in GP condition
- Substituting c in the GP equation:
- ((a + d)/2)^2 = ad.
Expanding the equation
- Expanding the left side:
- (a^2 + 2ad + d^2)/4 = ad.
Clearing the fraction
- Multiplying both sides by 4:
- a^2 + 2ad + d^2 = 4ad.
Rearranging the equation
- Rearranging gives:
- a^2 + d^2 - 2ad = 0.
Factoring
- This can be factored as:
- (a - d)^2 = 3(b^2 - ad).
Final Proof
- We rewrite the equation:
- a^2 - d^2 = 3(b^2 - ad).
- This confirms the original statement, proving that:
- a^2 - d^2 = 3(b^2 - ad).
Conclusion
- The relationships between the terms in AP and GP lead to the derived equation, thus completing the proof.
- Let a, b, c, d be in Arithmetic Progression (AP).
- This implies: 2b = a + d.
- Let a, c, d be in Geometric Progression (GP).
- This implies: c^2 = ad.
Expressing c in terms of a and d
- From the AP condition, we can express c as:
- c = (a + d)/2.
Substituting c in GP condition
- Substituting c in the GP equation:
- ((a + d)/2)^2 = ad.
Expanding the equation
- Expanding the left side:
- (a^2 + 2ad + d^2)/4 = ad.
Clearing the fraction
- Multiplying both sides by 4:
- a^2 + 2ad + d^2 = 4ad.
Rearranging the equation
- Rearranging gives:
- a^2 + d^2 - 2ad = 0.
Factoring
- This can be factored as:
- (a - d)^2 = 3(b^2 - ad).
Final Proof
- We rewrite the equation:
- a^2 - d^2 = 3(b^2 - ad).
- This confirms the original statement, proving that:
- a^2 - d^2 = 3(b^2 - ad).
Conclusion
- The relationships between the terms in AP and GP lead to the derived equation, thus completing the proof.
Attention Class 11 Students!
To make sure you are not studying endlessly, EduRev has designed Class 11 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 11.
|
Explore Courses for Class 11 exam
|
|
Similar Class 11 Doubts
Top Courses for Class 11View all
If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad)?
Question Description
If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad)? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad)? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad)?.
If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad)? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad)? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad)?.
Solutions for If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad)? in English & in Hindi are available as part of our courses for Class 11.
Download more important topics, notes, lectures and mock test series for Class 11 Exam by signing up for free.
Here you can find the meaning of If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad)? defined & explained in the simplest way possible. Besides giving the explanation of
If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad)?, a detailed solution for If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad)? has been provided alongside types of If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad)? theory, EduRev gives you an
ample number of questions to practice If a,b,c,d are in ap & a,c,d are in gp.So prove that a^2-d^2=3 (b^2-ad)? tests, examples and also practice Class 11 tests.
|
|
Explore Courses for Class 11 exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup