The equation of the plane through the line `(x-1)/2=(y-2)/3=(z-3)/4` and parallel to the co-ordinate axes are ….....

Answer

To find the equation of the plane that passes through the given line and is parallel to the coordinate axes, we can follow these steps:

### Step 1: Understand the given line
The line is represented in symmetric form as:
\[ \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} \]
This indicates that the line passes through the point \( (1, 2, 3) \) and has direction ratios \( (2, 3, 4) \).

### Step 2: Identify the plane's characteristics
Since the plane is parallel to the coordinate axes, it will not be inclined towards any of the axes. This means that the plane will have a constant value for one of the coordinates (x, y, or z) while the other two can vary.

### Step 3: Determine the equations of the planes
1. **Plane parallel to the yz-plane**: This plane will have a constant x-coordinate. Since the line passes through the point \( (1, 2, 3) \), the equation of the plane is:
\[ x = 1 \]

2. **Plane parallel to the xz-plane**: This plane will have a constant y-coordinate. Again, using the point \( (1, 2, 3) \), the equation of this plane is:
\[ y = 2 \]

3. **Plane parallel to the xy-plane**: This plane will have a constant z-coordinate. Using the point \( (1, 2, 3) \) once more, the equation of this plane is:
\[ z = 3 \]

### Final Answer
Thus, the equations of the planes through the line and parallel to the coordinate axes are:
1. \( x = 1 \)
2. \( y = 2 \)
3. \( z = 3 \)