Let the equations of a line and plane be `(x+3)/2=(y-4)/3=(z+5)/2a n d4x-2y-z=1,` respectively, then a. the line is parallel to the plane b. the line is perpendicular to the plane c. the line lies in the plane d. none of these

Options

A. the line is parallel to the plane
B. the line is parpendicular to the plane
C. the line lies in the plane
D. none of these

Explanation

To determine the relationship between the given line and the plane, we need to analyze their equations step by step.

### Step 1: Identify the equations of the line and the plane

The equation of the line is given in symmetric form:
\[ \frac{x + 3}{2} = \frac{y - 4}{3} = \frac{z + 5}{2} \]
The equation of the plane is given as:
\[ 4x - 2y - z = 1 \]

### Step 2: Extract the direction ratios of the line

From the symmetric equation of the line, we can identify:
- The direction ratios of the line are \( (2, 3, 2) \).

### Step 3: Identify a point on the line

To find a point on the line, we can set the parameter equal to zero:
- When \( t = 0 \):
\[ x = -3, \quad y = 4, \quad z = -5 \]
Thus, the point on the line is \( (-3, 4, -5) \).

### Step 4: Determine the normal vector of the plane

The normal vector to the plane can be derived from the coefficients of \( x, y, z \) in the plane equation:
- The normal vector \( \vec{n} \) is given by \( (4, -2, -1) \).

### 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. This can be checked using the dot product:
\[ \vec{d} \cdot \vec{n} = 2 \cdot 4 + 3 \cdot (-2) + 2 \cdot (-1) \]
Calculating this gives:
\[ = 8 - 6 - 2 = 0 \]
Since the dot product is zero, the line is parallel to the plane.

### Step 6: Check if the line is perpendicular to the plane

For the line to be perpendicular to the plane, the direction ratios of the line must be proportional to the normal vector of the plane. This is not the case here, as:
\[ (2, 3, 2) \text{ is not proportional to } (4, -2, -1). \]
Thus, the line is not perpendicular to the plane.

### Step 7: Check if the line lies in the plane

To check if the line lies in the plane, we substitute the point \( (-3, 4, -5) \) into the plane equation:
\[ 4(-3) - 2(4) - (-5) = -12 - 8 + 5 = -15 \neq 1 \]
Since the left-hand side does not equal the right-hand side, the line does not lie in the plane.

### Conclusion

Based on the analysis:
- The line is parallel to the plane.

### Final Answer
The correct option is:
**a. the line is parallel to the plane.**

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