JEE Exam > JEE Questions > If 2y - a is the harmonic mean between y − ...
Start Learning for Free
If 2y - a is the harmonic mean between y − x and y − z , then x - a, y - a and z - a are in
- a)arithmetic progression
- b)geometric progression
- c)harmonic progression
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
If 2y - a is the harmonic mean between y − x and y − z , then x - a,...
Given information:
The harmonic mean between two numbers is the reciprocal of the arithmetic mean of their reciprocals.
Let's break down the given information and solve the problem step by step.
Step 1: Write the given equation.
2y - a = Harmonic mean between (y - x) and (y - z)
Step 2: Write the definition of harmonic mean.
The harmonic mean between two numbers (a and b) is given by:
Harmonic mean = 2/(1/a + 1/b)
Step 3: Apply the definition of harmonic mean.
We can rewrite the given equation as:
2y - a = 2/(1/(y - x) + 1/(y - z))
Step 4: Simplify the equation.
To simplify the equation, we need to find a common denominator for the fractions on the right side:
2y - a = 2/((y - x + y - z)/(y - x)(y - z))
To divide by a fraction, we can multiply by its reciprocal:
2y - a = 2*(y - x)(y - z)/(2y - (x + z))
Step 5: Further simplify the equation.
2y - a = (y - x)(y - z)/(y - (x + z)/2)
Step 6: Simplify the right side of the equation.
2y - a = (y - x)(y - z)/(y - (x + z)/2)
2y - a = (y - x)(y - z)/(y - (x + z)/2)
2y - a = (y - x)(y - z)/(2y - x - z)
Step 7: Expand the right side of the equation.
2y - a = (y^2 - yz - xy + xz)/(2y - x - z)
Step 8: Multiply both sides by (2y - x - z) to eliminate the denominator.
(2y - a)*(2y - x - z) = y^2 - yz - xy + xz
Step 9: Expand and simplify the left side of the equation.
4y^2 - 2ay - 2xy + ax + 2yz - az - 2yz + xz - az = y^2 - yz - xy + xz
Step 10: Combine like terms.
3y^2 - 2ay - 2xy + ax - az - yz = y^2 - yz - xy + xz
Step 11: Move all terms to one side of the equation.
2y^2 - 2ay - 2xy + ax - az - yz - y^2 + yz + xy - xz = 0
Step 12: Simplify the left side of the equation.
y^2 - 2ay - 2xy + ax - az - xz = 0
Step 13: Rearrange the terms.
y^2 - 2ay - 2xy + ax - az - xz = 0
The harmonic mean between two numbers is the reciprocal of the arithmetic mean of their reciprocals.
Let's break down the given information and solve the problem step by step.
Step 1: Write the given equation.
2y - a = Harmonic mean between (y - x) and (y - z)
Step 2: Write the definition of harmonic mean.
The harmonic mean between two numbers (a and b) is given by:
Harmonic mean = 2/(1/a + 1/b)
Step 3: Apply the definition of harmonic mean.
We can rewrite the given equation as:
2y - a = 2/(1/(y - x) + 1/(y - z))
Step 4: Simplify the equation.
To simplify the equation, we need to find a common denominator for the fractions on the right side:
2y - a = 2/((y - x + y - z)/(y - x)(y - z))
To divide by a fraction, we can multiply by its reciprocal:
2y - a = 2*(y - x)(y - z)/(2y - (x + z))
Step 5: Further simplify the equation.
2y - a = (y - x)(y - z)/(y - (x + z)/2)
Step 6: Simplify the right side of the equation.
2y - a = (y - x)(y - z)/(y - (x + z)/2)
2y - a = (y - x)(y - z)/(y - (x + z)/2)
2y - a = (y - x)(y - z)/(2y - x - z)
Step 7: Expand the right side of the equation.
2y - a = (y^2 - yz - xy + xz)/(2y - x - z)
Step 8: Multiply both sides by (2y - x - z) to eliminate the denominator.
(2y - a)*(2y - x - z) = y^2 - yz - xy + xz
Step 9: Expand and simplify the left side of the equation.
4y^2 - 2ay - 2xy + ax + 2yz - az - 2yz + xz - az = y^2 - yz - xy + xz
Step 10: Combine like terms.
3y^2 - 2ay - 2xy + ax - az - yz = y^2 - yz - xy + xz
Step 11: Move all terms to one side of the equation.
2y^2 - 2ay - 2xy + ax - az - yz - y^2 + yz + xy - xz = 0
Step 12: Simplify the left side of the equation.
y^2 - 2ay - 2xy + ax - az - xz = 0
Step 13: Rearrange the terms.
y^2 - 2ay - 2xy + ax - az - xz = 0
Attention JEE Students!
To make sure you are not studying endlessly, EduRev has designed JEE study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in JEE.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
If 2y - a is the harmonic mean between y − x and y − z , then x - a, y - a and z - a are ina)arithmetic progressionb)geometric progressionc)harmonic progressiond)none of theseCorrect answer is option 'B'. Can you explain this answer?
Question Description
If 2y - a is the harmonic mean between y − x and y − z , then x - a, y - a and z - a are ina)arithmetic progressionb)geometric progressionc)harmonic progressiond)none of theseCorrect answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If 2y - a is the harmonic mean between y − x and y − z , then x - a, y - a and z - a are ina)arithmetic progressionb)geometric progressionc)harmonic progressiond)none of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 2y - a is the harmonic mean between y − x and y − z , then x - a, y - a and z - a are ina)arithmetic progressionb)geometric progressionc)harmonic progressiond)none of theseCorrect answer is option 'B'. Can you explain this answer?.
If 2y - a is the harmonic mean between y − x and y − z , then x - a, y - a and z - a are ina)arithmetic progressionb)geometric progressionc)harmonic progressiond)none of theseCorrect answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If 2y - a is the harmonic mean between y − x and y − z , then x - a, y - a and z - a are ina)arithmetic progressionb)geometric progressionc)harmonic progressiond)none of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 2y - a is the harmonic mean between y − x and y − z , then x - a, y - a and z - a are ina)arithmetic progressionb)geometric progressionc)harmonic progressiond)none of theseCorrect answer is option 'B'. Can you explain this answer?.
Solutions for If 2y - a is the harmonic mean between y − x and y − z , then x - a, y - a and z - a are ina)arithmetic progressionb)geometric progressionc)harmonic progressiond)none of theseCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of If 2y - a is the harmonic mean between y − x and y − z , then x - a, y - a and z - a are ina)arithmetic progressionb)geometric progressionc)harmonic progressiond)none of theseCorrect answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If 2y - a is the harmonic mean between y − x and y − z , then x - a, y - a and z - a are ina)arithmetic progressionb)geometric progressionc)harmonic progressiond)none of theseCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for If 2y - a is the harmonic mean between y − x and y − z , then x - a, y - a and z - a are ina)arithmetic progressionb)geometric progressionc)harmonic progressiond)none of theseCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of If 2y - a is the harmonic mean between y − x and y − z , then x - a, y - a and z - a are ina)arithmetic progressionb)geometric progressionc)harmonic progressiond)none of theseCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If 2y - a is the harmonic mean between y − x and y − z , then x - a, y - a and z - a are ina)arithmetic progressionb)geometric progressionc)harmonic progressiond)none of theseCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE 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