If x,2y,3z are in A.P where the distinct numbers x,y,z are in G.P then the common ratio of G.P is

Options

A. 3
B. `1/3`
C. 2
D. 1

Answer

To solve the problem, we need to establish the relationships between the variables given that \( x, 2y, 3z \) are in Arithmetic Progression (A.P.) and \( x, y, z \) are in Geometric Progression (G.P.).

### Step-by-Step Solution

1. **Understanding A.P. Condition**:
Since \( x, 2y, 3z \) are in A.P., we can use the property of A.P. which states that the middle term is the average of the other two terms. Thus, we have:
\[ 2y = \frac{x + 3z}{2} \]
Multiplying both sides by 2 gives:
\[ 4y = x + 3z \quad \text{(Equation 1)} \]

2. **Understanding G.P. Condition**:
Since \( x, y, z \) are in G.P., the middle term \( y \) is the geometric mean of \( x \) and \( z \). Therefore, we can write:
\[ y^2 = xz \quad \text{(Equation 2)} \]

3. **Expressing y in terms of x and z**:
From Equation 2, we can express \( y \) as:
\[ y = \sqrt{xz} \]

4. **Substituting y in Equation 1**:
Substitute \( y \) from the previous step into Equation 1:
\[ 4\sqrt{xz} = x + 3z \]

5. **Isolating terms**:
To eliminate the square root, we can square both sides:
\[ (4\sqrt{xz})^2 = (x + 3z)^2 \]
This simplifies to:
\[ 16xz = x^2 + 6xz + 9z^2 \]

6. **Rearranging the equation**:
Rearranging gives:
\[ 16xz - 6xz - 9z^2 - x^2 = 0 \]
Simplifying further:
\[ 10xz - x^2 - 9z^2 = 0 \]

7. **Rearranging into a standard quadratic form**:
Rearranging gives:
\[ x^2 - 10xz + 9z^2 = 0 \]

8. **Using the quadratic formula**:
We can solve for \( x \) using the quadratic formula:
\[ x = \frac{10z \pm \sqrt{(10z)^2 - 4 \cdot 1 \cdot 9z^2}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{10z \pm \sqrt{100z^2 - 36z^2}}{2} \]
\[ x = \frac{10z \pm \sqrt{64z^2}}{2} \]
\[ x = \frac{10z \pm 8z}{2} \]

9. **Finding the values of x**:
This gives us two possible values for \( x \):
\[ x = \frac{18z}{2} = 9z \quad \text{or} \quad x = \frac{2z}{2} = z \]

10. **Finding the common ratio**:
If \( x = 9z \), then the common ratio \( r \) can be expressed as:
\[ r = \frac{y}{x} = \frac{\sqrt{xz}}{x} = \frac{\sqrt{9z^2}}{9z} = \frac{3z}{9z} = \frac{1}{3} \]
If \( x = z \), we would have \( r = 1 \), but since \( x, y, z \) are distinct, we discard this case.

### Conclusion
Thus, the common ratio of the G.P. is:
\[ \boxed{\frac{1}{3}} \]