Consider the line `L-=(x-1)/(2)=(y+2)/(3)=(z-7)/(6)`. Point `P(2, -5, 0)` and Q are such that PQ is perpendicular to the line L and the midpoint of PQ lies on line L, then coordinates of Q are

Options

A. `(-4, -5, 2)`
B. `(-3, 0, 1)`
C. (1, 6, 2)
D. (1, 5, 7)

Answer

To solve the problem, we need to find the coordinates of point Q such that the line segment PQ is perpendicular to line L, and the midpoint of PQ lies on line L. Let's break this down step by step.

### Step 1: Understand the line L
The line L is given in symmetric form:
\[ \frac{x-1}{2} = \frac{y+2}{3} = \frac{z-7}{6} \]
We can express the coordinates of any point on line L in terms of a parameter \( \lambda \):
- \( x = 2\lambda + 1 \)
- \( y = 3\lambda - 2 \)
- \( z = 6\lambda + 7 \)

### Step 2: Define the coordinates of point P and point Q
Point P is given as \( P(2, -5, 0) \). Let the coordinates of point Q be \( Q(x_1, y_1, z_1) \).

### Step 3: Find the midpoint M of PQ
The midpoint M of segment PQ can be calculated as:
\[ M = \left( \frac{2 + x_1}{2}, \frac{-5 + y_1}{2}, \frac{0 + z_1}{2} \right) \]

### Step 4: Set the midpoint M on line L
Since M lies on line L, we can set:
\[ \frac{2 + x_1}{2} = 2\lambda + 1, \quad \frac{-5 + y_1}{2} = 3\lambda - 2, \quad \frac{z_1}{2} = 6\lambda + 7 \]

### Step 5: Find the direction ratios of PQ
The direction ratios of line L are \( (2, 3, 6) \). The direction ratios of PQ can be expressed as:
\[ (x_1 - 2, y_1 + 5, z_1 - 0) = (x_1 - 2, y_1 + 5, z_1) \]

### Step 6: Use the condition of perpendicularity
For PQ to be perpendicular to line L, the dot product of their direction ratios must equal zero:
\[ 2(x_1 - 2) + 3(y_1 + 5) + 6(z_1) = 0 \]

### Step 7: Substitute the expressions for \( x_1, y_1, z_1 \)
From the equations of M, we can express \( x_1, y_1, z_1 \) in terms of \( \lambda \):
1. From \( \frac{2 + x_1}{2} = 2\lambda + 1 \):
\[ x_1 = 4\lambda + 2 - 2 = 4\lambda \]
2. From \( \frac{-5 + y_1}{2} = 3\lambda - 2 \):
\[ y_1 = 6\lambda - 4 + 5 = 6\lambda + 1 \]
3. From \( \frac{z_1}{2} = 6\lambda + 7 \):
\[ z_1 = 12\lambda + 14 \]

### Step 8: Substitute into the dot product equation
Substituting \( x_1, y_1, z_1 \) into the dot product equation:
\[ 2(4\lambda - 2) + 3(6\lambda + 1 + 5) + 6(12\lambda + 14) = 0 \]
Simplifying:
\[ 8\lambda - 4 + 18\lambda + 18 + 72\lambda + 84 = 0 \]
Combining like terms:
\[ (8 + 18 + 72)\lambda + (84 + 18 - 4) = 0 \]
\[ 98\lambda + 98 = 0 \implies \lambda = -1 \]

### Step 9: Find coordinates of Q
Substituting \( \lambda = -1 \) back into the equations for \( x_1, y_1, z_1 \):
1. \( x_1 = 4(-1) = -4 \)
2. \( y_1 = 6(-1) + 1 = -5 \)
3. \( z_1 = 12(-1) + 14 = 2 \)

Thus, the coordinates of point Q are:
\[ Q(-4, -5, 2) \]

### Final Answer
The coordinates of Q are \( (-4, -5, 2) \).