For the `l:(x-1)/3=(y+1)/2=(z-3)/(-1)` and the plane `P:x-2y-z=0` of the following assertions the ony one which is true is (A) l lies in P (B) l is parallel to P (C) l is perpendiculr to P (D) none of these

Options

A. L is `bot ` to `pi`
B. L lies in `pi`
C. L is parallel to `pi`
D. none of these

Answer

To determine the relationship between the line \( l \) and the plane \( P \), we will analyze the given equations step by step.

### Step 1: Identify the Line and Plane Equations
The line \( l \) is given in symmetric form:
\[ \frac{x-1}{3} = \frac{y+1}{2} = \frac{z-3}{-1} \]
This can be rewritten in parametric form as:
\[ x = 1 + 3t, \quad y = -1 + 2t, \quad z = 3 - t \]
where \( t \) is a parameter.

The plane \( P \) is given by the equation:
\[ x - 2y - z = 0 \]

### Step 2: Find Direction Ratios of the Line
From the parametric equations, the direction ratios of the line \( l \) are:
\[ (3, 2, -1) \]

### Step 3: Find Normal Vector of the Plane
The coefficients of the plane equation \( x - 2y - z = 0 \) give us the normal vector of the plane:
\[ (1, -2, -1) \]

### Step 4: Check if the Line Lies in the Plane
To check if the line lies in the plane, we can substitute a point from the line into the plane equation. Let's use the point when \( t = 0 \):
\[ x = 1, \quad y = -1, \quad z = 3 \]
Now substitute these values into the plane equation:
\[ 1 - 2(-1) - 3 = 1 + 2 - 3 = 0 \]
Since the equation holds true, the point \( (1, -1, 3) \) lies on the plane \( P \).

### Step 5: Check if the Line is Parallel to the Plane
For the line to be parallel to the plane, the direction ratios of the line must be perpendicular to the normal vector of the plane. We check this by taking the dot product of the direction ratios of the line and the normal vector of the plane:
\[ (3, 2, -1) \cdot (1, -2, -1) = 3 \cdot 1 + 2 \cdot (-2) + (-1) \cdot (-1) = 3 - 4 + 1 = 0 \]
Since the dot product is zero, the line is perpendicular to the plane, which means it is not parallel.

### Step 6: Check if the Line is Perpendicular to the Plane
Since the dot product calculated in the previous step is zero, it confirms that the line is perpendicular to the plane.

### Conclusion
From the analysis, we have determined that:
- The line \( l \) lies in the plane \( P \).
- The line is also perpendicular to the plane.

Thus, the correct assertion is:
**(A) \( l \) lies in \( P \)**.