If x,y,z, are in A.P. and `x^(2),y^(2),z^(2)` are in H.P., then
which of the following is correct ?

Options

A. x,y,z are all equal
B. `-(x)/(2),y,z` are G.P.
C. `(x)/(2),y,z` are in G.P.
D. Either (1) or(2)

Answer

To solve the problem, we need to analyze the conditions given: \(x, y, z\) are in Arithmetic Progression (A.P.) and \(x^2, y^2, z^2\) are in Harmonic Progression (H.P.).

### Step 1: Understanding A.P. Condition
Since \(x, y, z\) are in A.P., we can express this as:
\[ 2y = x + z \]
This implies:
\[ x + z = 2y \quad \text{(Equation 1)} \]

### Step 2: Understanding H.P. Condition
For \(x^2, y^2, z^2\) to be in H.P., the reciprocals \( \frac{1}{x^2}, \frac{1}{y^2}, \frac{1}{z^2} \) must be in A.P. This gives us:
\[ 2 \cdot \frac{1}{y^2} = \frac{1}{x^2} + \frac{1}{z^2} \]
Multiplying through by \(y^2 x^2 z^2\) to eliminate the denominators, we have:
\[ 2x^2 z^2 = z^2 + x^2 \]
Rearranging gives:
\[ 2y^2 = \frac{x^2 + z^2}{x^2 z^2} \quad \text{(Equation 2)} \]

### Step 3: Expressing \(x^2 + z^2\)
From Equation 1, we can express \(x^2 + z^2\) in terms of \(y\):
\[ x^2 + z^2 = (x + z)^2 - 2xz = (2y)^2 - 2xz = 4y^2 - 2xz \]

### Step 4: Substituting into H.P. Condition
Substituting \(x^2 + z^2\) into Equation 2:
\[ 2y^2 = \frac{4y^2 - 2xz}{xz} \]
Cross-multiplying gives:
\[ 2y^2 xz = 4y^2 - 2xz \]
Rearranging leads to:
\[ 2y^2 xz + 2xz = 4y^2 \]
Factoring out \(2xz\):
\[ 2xz(y^2 + 1) = 4y^2 \]
Dividing both sides by 2:
\[ xz(y^2 + 1) = 2y^2 \]

### Step 5: Analyzing Options
Now we check the given options:

1. **Option 1: \(x = y = z\)**
If \(x = y = z\), then \(y^2 + 1 = 2\) and \(xz = y^2\). This holds true.

2. **Option 2: \(-\frac{x}{2}, y, z\)**
If we substitute and check, we find that this also holds true.

3. **Option 3: \(\frac{x}{2}, y, z\)**
Substituting this does not satisfy the condition.

### Conclusion
Thus, the correct options are **Option 1 and Option 2**.