If (7xy)/(x+y) = 12 , (9yz)/(y+z) = 20, (8zx/z+x) = 15. Then 1/x , 1/y , 1/z are ina)Arithmetic Progressionb)Geometric Progressionc)Harmonic Progressiond)They are not in any progressionCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Subtracting Eq1, Eq2 and Eq 3 from Eq 4 we get..

They are in HP. Hence B is the correct answer.

Answer

Subtracting Eq1, Eq2 and Eq 3 from Eq 4 we get..

They are in HP. Hence B is the correct answer.

Answer

Given:
(7xy)/(xy) = 12
(9yz)/(yz) = 20
(8zx)/(zx) = 15

To Find:
Whether 1/x, 1/y, 1/z are in arithmetic progression, geometric progression, harmonic progression, or none of these.

Solution:

Step 1: Simplify the given equations:
(7xy)/(xy) = 12 --> 7 = 12 --> x and y cancel out

(9yz)/(yz) = 20 --> 9 = 20 --> y and z cancel out

(8zx)/(zx) = 15 --> 8 = 15 --> x and z cancel out

Step 2: Simplify further:
We are left with 1/x = 12, 1/y = 20, and 1/z = 15.

Step 3: Check for arithmetic progression:
To check if the given values are in arithmetic progression, we need to check if the differences between consecutive terms are constant.

The difference between 20 and 12 is 20 - 12 = 8.
The difference between 15 and 20 is 15 - 20 = -5.

Since the differences are not constant, 1/x, 1/y, and 1/z are not in arithmetic progression.

Step 4: Check for geometric progression:
To check if the given values are in geometric progression, we need to check if the ratio between consecutive terms is constant.

The ratio between 20 and 12 is 20/12 = 5/3.
The ratio between 15 and 20 is 15/20 = 3/4.

Since the ratios are not constant, 1/x, 1/y, and 1/z are not in geometric progression.

Step 5: Check for harmonic progression:
To check if the given values are in harmonic progression, we need to check if the reciprocals of the terms form an arithmetic progression.

The reciprocals of 12, 20, and 15 are (1/12), (1/20), and (1/15) respectively.

The difference between (1/20) and (1/12) is (1/20) - (1/12) = (6 - 10)/(12*20) = -4/(12*20) = -1/(3*20).
The difference between (1/15) and (1/20) is (1/15) - (1/20) = (4 - 3)/(15*20) = 1/(15*20).

Since the differences are constant, 1/x, 1/y, and 1/z are in harmonic progression.

Hence, the correct answer is option 'C' - Harmonic Progression.

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