If X,Y and Z are three points such that XY = 2YZ and XZ =3YZ , then the three points are :

Options

A. not collinear
B. collinear and X lies between Y and Z
C. collinear and Y lies between X and Z
D. collinear and Z lies between X and Y

Answer

To solve the problem, we need to analyze the relationships between the points X, Y, and Z based on the given conditions:

1. **Given Relationships**:
- \( XY = 2 \times YZ \)
- \( XZ = 3 \times YZ \)

2. **Expressing in Terms of YZ**:
- Let \( YZ = d \) (where d is a positive distance).
- Then, from the first relationship:
\[ XY = 2d \]
- From the second relationship:
\[ XZ = 3d \]

3. **Understanding the Points**:
- We can visualize the points on a line:
- Start from point Y, and move to point Z. The distance \( YZ \) is \( d \).
- Since \( XY = 2d \), point X must be located at a distance of \( 2d \) from point Y.
- Since \( XZ = 3d \), point X must also be located at a distance of \( 3d \) from point Z.

4. **Positioning the Points**:
- Place point Y at the origin (0).
- Then, point Z will be at \( d \) (i.e., \( YZ = d \)).
- Now, point X, which is \( 2d \) from Y, will be at \( 2d \).

5. **Verifying the Position**:
- The distance \( XZ \) can be calculated:
\[ XZ = X - Z = 2d - d = d \]
- However, we need to check the relationship \( XZ = 3d \). This means we need to adjust our understanding:
- If we consider the distances correctly, we see that:
- \( XZ = 3d \) implies that point X is actually positioned such that it is 3 times the distance from Z, which means it must be further away from Z.

6. **Final Arrangement**:
- Therefore, if we place Y at 0, Z at \( d \), and X at \( 3d \), we can confirm:
- \( XY = 2d \) (from Y to X)
- \( XZ = 3d \) (from Z to X)
- Thus, the points are indeed collinear and positioned as follows:
- \( Y \) at 0, \( Z \) at \( d \), and \( X \) at \( 3d \).

### Conclusion:
The three points X, Y, and Z are collinear, with the following relationships:
- \( Y \) is at 0,
- \( Z \) is at \( d \),
- \( X \) is at \( 3d \).