If ` 2 (y - a) ` is the ` H.M.` between ` y - x and y - z ` then ` x-a, y-a, z-a` are in

Options

A. arithmetic progression
B. geometri progression
C. harmonic progression
D. none of these

Answer

To solve the problem, we need to show that if \( 2(y - a) \) is the harmonic mean between \( y - x \) and \( y - z \), then \( x - a, y - a, z - a \) are in geometric progression (G.P.).

### Step 1: Understanding the Harmonic Mean
The harmonic mean \( H \) of two numbers \( a \) and \( b \) is given by the formula:
\[ H = \frac{2ab}{a + b} \]
In our case, we have:
\[ H = 2(y - a) \quad \text{and} \quad a = y - x, \quad b = y - z \]

### Step 2: Setting Up the Equation
According to the definition of the harmonic mean, we can set up the equation:
\[ 2(y - a) = \frac{2(y - x)(y - z)}{(y - x) + (y - z)} \]
Simplifying the denominator:
\[ (y - x) + (y - z) = 2y - (x + z) \]
Thus, we rewrite the equation as:
\[ 2(y - a)(2y - (x + z)) = 2(y - x)(y - z) \]

### Step 3: Cross Multiplying
Cross multiplying gives us:
\[ 2(y - a)(2y - (x + z)) = 2(y - x)(y - z) \]
This simplifies to:
\[ 2(y - a)(2y - x - z) = 2(y^2 - (x + z)y + xz) \]

### Step 4: Expanding Both Sides
Expanding both sides:
\[ 2(2y^2 - 2ay - xy + ax - zy + az) = 2y^2 - 2xy - 2zy + 2xz \]

### Step 5: Rearranging Terms
Rearranging the equation leads to:
\[ 4y^2 - 4ay - 2xy + 2ax - 2zy + 2az = 2y^2 - 2xy - 2zy + 2xz \]
Combining like terms results in:
\[ 2y^2 - 4ay + 2ax + 2az - 2xz = 0 \]

### Step 6: Completing the Square
To complete the square, we can rewrite the equation:
\[ (y - a)^2 = (x - a)(z - a) \]

### Conclusion: Identifying the Progression
From the equation \( (y - a)^2 = (x - a)(z - a) \), we can conclude that \( y - a \) is the geometric mean of \( x - a \) and \( z - a \). Therefore, \( x - a, y - a, z - a \) are in geometric progression.

### Final Answer
Thus, \( x - a, y - a, z - a \) are in G.P.