If ` 2 (y - a) ` is the ` H.M.` between ` y - x and y - z ` then ` x-a, y-a, z-a` are in (i) A.P (ii) G.P (iii) H.P (iv) none of these

Options

A. A.P.
B. G.P.
C. H.P.
D. none of these

Answer

To solve the problem step by step, we start with the given information that \( 2(y - a) \) is the harmonic mean between \( y - x \) and \( y - z \).

### 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:
\[ H = 2(y - a), \quad a = y - x, \quad b = y - z \]

### Step 2: Setting Up the Equation
According to the definition of harmonic mean, we can set up the equation:
\[ 2(y - a) = \frac{2(y - x)(y - z)}{(y - x) + (y - z)} \]
This simplifies to:
\[ 2(y - a) = \frac{2(y - x)(y - z)}{2y - (x + z)} \]
We can cancel the 2 from both sides:
\[ y - a = \frac{(y - x)(y - z)}{2y - (x + z)} \]

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

### Step 4: Expanding Both Sides
Expanding both sides, we have:
\[ 2y(y - a) - (x + z)(y - a) = (y^2 - yx - yz + xz) \]
This leads to:
\[ 2y^2 - 2ay - xy + ax - zy + az = y^2 - yx - yz + xz \]

### Step 5: Rearranging the Equation
Rearranging the equation gives us:
\[ 2y^2 - y^2 - 2ay + ax + az = xz - yz \]
This simplifies to:
\[ y^2 - 2ay + ax + az = xz - yz \]

### Step 6: Factoring
Factoring out common terms, we can express the left side as:
\[ y^2 - 2ay + ax + az = (y - a)^2 + a(x + z - 2a) \]
And the right side can be rearranged to:
\[ xz - az = z(x - a) \]

### Step 7: Conclusion
We can now see that:
\[ (y - a)^2 = z(x - a) \]
This implies that \( x - a, y - a, z - a \) are in geometric progression (G.P.).

### Final Answer
Thus, the answer is:
**(ii) G.P.**