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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?
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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
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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?
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