Given that `log_(x)y,log_(z)x,log_(y)z` are in GP, xyz=64 and `x^(3),y^(3),z^(3)` are in AP.
Q. Which one of the following is correct ? x,y and z are

Options

A. In AP only
B. In GP only
C. In both AP and GP
D. Neither in AP nor in GP

Answer

To solve the problem step by step, we will analyze the given conditions and derive the necessary relationships.

### Step 1: Understand the Given Information
We have three logarithmic expressions:
1. \( \log_x y \)
2. \( \log_z x \)
3. \( \log_y z \)

These are in geometric progression (GP). Additionally, we know that \( xyz = 64 \) and \( x^3, y^3, z^3 \) are in arithmetic progression (AP).

### Step 2: Use the GP Condition
For three numbers \( a, b, c \) to be in GP, the condition is:
\[ b^2 = ac \]
Applying this to our logarithmic expressions:
\[ (\log_z x)^2 = \log_x y \cdot \log_y z \]

### Step 3: Rewrite the Logarithms
Using the change of base formula:
- \( \log_z x = \frac{\log x}{\log z} \)
- \( \log_x y = \frac{\log y}{\log x} \)
- \( \log_y z = \frac{\log z}{\log y} \)

Substituting these into the GP condition:
\[ \left(\frac{\log x}{\log z}\right)^2 = \left(\frac{\log y}{\log x}\right) \cdot \left(\frac{\log z}{\log y}\right) \]

### Step 4: Simplify the GP Condition
This simplifies to:
\[ \frac{(\log x)^2}{(\log z)^2} = \frac{\log z}{\log x} \]
Cross-multiplying gives:
\[ (\log x)^3 = (\log z)^2 \]

### Step 5: Relate \( x \) and \( z \)
Taking the cube root on both sides, we find:
\[ \log x = \log z \implies x = z \]

### Step 6: Use the AP Condition
Now, since \( x^3, y^3, z^3 \) are in AP, we have:
\[ 2y^3 = x^3 + z^3 \]
Since \( x = z \), we can substitute:
\[ 2y^3 = x^3 + x^3 = 2x^3 \implies y^3 = x^3 \implies y = x \]

### Step 7: Conclude Relationships
From the above steps, we have:
\[ x = y = z \]

### Step 8: Use the Product Condition
Given that \( xyz = 64 \):
\[ x \cdot x \cdot x = 64 \implies x^3 = 64 \implies x = 4 \]
Thus, \( x = y = z = 4 \).

### Final Conclusion
Since \( x, y, z \) are equal, they are in both AP and GP.

### Answer
The correct option is that \( x, y, z \) are in both AP and GP.

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