To solve the problem step by step, we will follow the reasoning provided in the video transcript.
### Step 1: Identify the Points
We have three points through which the plane \( P \) passes:
1. \( A(2, 1, 0) \)
2. \( B(\lambda - 1, 1, 1) \)
3. \( C(\lambda, 0, 1) \)
We also have two points:
- Point \( R(2, 1, 6) \)
- Point \( Q(6, 5, -2) \), which is the image of point \( R \) in the plane \( P \).
### Step 2: Find the Vector \( \overrightarrow{RQ} \)
To find the vector \( \overrightarrow{RQ} \), we subtract the coordinates of point \( R \) from point \( Q \):
\[
\overrightarrow{RQ} = Q - R = (6 - 2, 5 - 1, -2 - 6) = (4, 4, -8)
\]
### Step 3: Define a Point on the Plane
Let \( P(x, y, z) \) be any point on the plane. We can express the vector from point \( P \) to point \( A \) as:
\[
\overrightarrow{XP} = (x - 2, y - 1, z - 0) = (x - 2, y - 1, z)
\]
### Step 4: Use the Perpendicular Condition
Since \( \overrightarrow{RQ} \) is perpendicular to the plane, the dot product of \( \overrightarrow{RQ} \) and \( \overrightarrow{XP} \) must equal zero:
\[
\overrightarrow{RQ} \cdot \overrightarrow{XP} = 0
\]
This gives us:
\[
(4, 4, -8) \cdot (x - 2, y - 1, z) = 0
\]
Calculating the dot product:
\[
4(x - 2) + 4(y - 1) - 8z = 0
\]
Expanding this:
\[
4x - 8 + 4y - 4 - 8z = 0
\]
Simplifying:
\[
4x + 4y - 8z - 12 = 0
\]
### Step 5: Substitute Points into the Plane Equation
Now we will substitute the coordinates of points \( B \) and \( C \) into the plane equation to find \( \lambda \).
**Substituting Point \( B(\lambda - 1, 1, 1) \):**
\[
4(\lambda - 1) + 4(1) - 8(1) - 12 = 0
\]
This simplifies to:
\[
4\lambda - 4 + 4 - 8 - 12 = 0
\]
\[
4\lambda - 20 = 0 \implies \lambda = 5
\]
**Substituting Point \( C(\lambda, 0, 1) \):**
\[
4(\lambda) + 4(0) - 8(1) - 12 = 0
\]
This simplifies to:
\[
4\lambda - 8 - 12 = 0
\]
\[
4\lambda - 20 = 0 \implies \lambda = 5
\]
### Conclusion
In both cases, we find that \( \lambda = 5 \).
### Final Answer
Thus, the value of \( \lambda \) is \( 5 \).
