Let `P=-(1,7,sqrt(2))` be a point and line L is `2sqrt(2)(x-1)=y-2,z=0`. If PQ is the distance of plane `sqrt(2)x+y-z=1` from point P measured along a line inclined at an angle of `45^(@)` with the line L and is minimum then the value of PQ is

Options

A. 3
B. 4
C. 6
D. 8

Explanation

To solve the problem, we need to find the distance \( PQ \) from the point \( P = -(1, 7, \sqrt{2}) \) to the plane given by the equation \( \sqrt{2}x + y - z = 1 \). This distance is measured along a line inclined at an angle of \( 45^\circ \) with the line \( L \), but since the problem states that this distance is minimum, we can directly calculate the perpendicular distance from the point to the plane.

### Step-by-Step Solution:

1. **Identify the components of the plane equation**:
The equation of the plane is given as:
\[ \sqrt{2}x + y - z = 1 \]
Here, we can identify \( a = \sqrt{2} \), \( b = 1 \), \( c = -1 \), and \( d = 1 \).

2. **Identify the coordinates of point \( P \)**:
The coordinates of point \( P \) are:
\[ P = (1, 7, \sqrt{2}) \]
Thus, \( x_1 = 1 \), \( y_1 = 7 \), and \( z_1 = \sqrt{2} \).

3. **Use the formula for the perpendicular distance from a point to a plane**:
The formula for the perpendicular distance \( D \) from a point \( (x_1, y_1, z_1) \) to a plane \( ax + by + cz = d \) is given by:
\[ D = \frac{|ax_1 + by_1 + cz_1 - d|}{\sqrt{a^2 + b^2 + c^2}} \]

4. **Substitute the values into the formula**:
Substitute \( a \), \( b \), \( c \), \( d \), \( x_1 \), \( y_1 \), and \( z_1 \) into the formula:
\[ D = \frac{|\sqrt{2} \cdot 1 + 1 \cdot 7 - 1 \cdot \sqrt{2} - 1|}{\sqrt{(\sqrt{2})^2 + 1^2 + (-1)^2}} \]

5. **Calculate the numerator**:
Simplifying the numerator:
\[ \sqrt{2} \cdot 1 + 7 - \sqrt{2} - 1 = 7 - 1 = 6 \]
So, the numerator becomes:
\[ |6| = 6 \]

6. **Calculate the denominator**:
Now, calculate the denominator:
\[ \sqrt{(\sqrt{2})^2 + 1^2 + (-1)^2} = \sqrt{2 + 1 + 1} = \sqrt{4} = 2 \]

7. **Calculate the distance \( D \)**:
Now, substitute the values back into the distance formula:
\[ D = \frac{6}{2} = 3 \]

Thus, the value of \( PQ \) is \( 3 \).

### Final Answer:
\[ PQ = 3 \]