To solve the problem, we need to find the coordinates of point Q given the conditions that PQ is perpendicular to the line L and the midpoint of PQ lies on line L.
### Step 1: Understand the line L
The line L is given in the symmetric form:
\[
\frac{x-1}{2} = \frac{y+2}{3} = \frac{z-7}{6}
\]
From this, we can extract the direction ratios of the line:
- Direction ratios: \( (2, 3, 6) \)
### Step 2: Define the coordinates of point Q
Let the coordinates of point Q be \( (α, β, γ) \). The coordinates of point P are given as \( P(2, -5, 0) \).
### Step 3: Find the direction ratios of line PQ
The direction ratios of line PQ can be expressed as:
\[
(α - 2, β + 5, γ - 0) = (α - 2, β + 5, γ)
\]
### Step 4: Use the condition of perpendicularity
For PQ to be perpendicular to line L, the dot product of their direction ratios must equal zero:
\[
2(α - 2) + 3(β + 5) + 6(γ) = 0
\]
Expanding this gives:
\[
2α - 4 + 3β + 15 + 6γ = 0
\]
Simplifying, we have:
\[
2α + 3β + 6γ + 11 = 0 \quad \text{(Equation 1)}
\]
### Step 5: Find the midpoint of PQ
The midpoint M of segment PQ can be calculated as:
\[
M = \left( \frac{2 + α}{2}, \frac{-5 + β}{2}, \frac{0 + γ}{2} \right)
\]
Since the midpoint M lies on line L, we can express the coordinates of M in terms of a parameter \( λ \):
\[
M = (2λ + 1, 3λ - 2, 6λ + 7)
\]
### Step 6: Set up equations for the midpoint
Equating the coordinates of M with the coordinates from line L gives us:
1. \( \frac{2 + α}{2} = 2λ + 1 \)
2. \( \frac{-5 + β}{2} = 3λ - 2 \)
3. \( \frac{γ}{2} = 6λ + 7 \)
From these equations, we can solve for \( α, β, γ \) in terms of \( λ \):
1. \( 2 + α = 4λ + 2 \) → \( α = 4λ \)
2. \( -5 + β = 6λ - 4 \) → \( β = 6λ + 1 \)
3. \( γ = 12λ + 14 \)
### Step 7: Substitute into Equation 1
Now substitute \( α, β, γ \) into Equation 1:
\[
2(4λ) + 3(6λ + 1) + 6(12λ + 14) + 11 = 0
\]
Expanding this gives:
\[
8λ + 18λ + 3 + 72λ + 84 + 11 = 0
\]
Combining like terms:
\[
98λ + 98 = 0
\]
Thus,
\[
λ = -1
\]
### Step 8: Find coordinates of Q
Substituting \( λ = -1 \) back into the equations for \( α, β, γ \):
1. \( α = 4(-1) = -4 \)
2. \( β = 6(-1) + 1 = -5 \)
3. \( γ = 12(-1) + 14 = 2 \)
### Final Answer
The coordinates of point Q are:
\[
Q(-4, -5, 2)
\]
