The equation of a plane which bisects the line joining `(1, 5, 7) and (-3, 1, -1)` is `x+y+2z=lambda`, then find `lambda`.

Answer

To solve the problem, we need to find the value of \( \lambda \) in the equation of the plane \( x + y + 2z = \lambda \) that bisects the line segment joining the points \( (1, 5, 7) \) and \( (-3, 1, -1) \).

### Step-by-Step Solution:

1. **Identify the Points:**
Let the first point be \( A(1, 5, 7) \) and the second point be \( B(-3, 1, -1) \).

2. **Find the Midpoint of the Line Segment:**
The midpoint \( M \) of the line segment joining points \( A \) and \( B \) can be calculated using the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]
Substituting the coordinates of points \( A \) and \( B \):
\[ M = \left( \frac{1 + (-3)}{2}, \frac{5 + 1}{2}, \frac{7 + (-1)}{2} \right) \]
Simplifying this gives:
\[ M = \left( \frac{-2}{2}, \frac{6}{2}, \frac{6}{2} \right) = (-1, 3, 3) \]

3. **Substitute Midpoint Coordinates into the Plane Equation:**
The equation of the plane is given by:
\[ x + y + 2z = \lambda \]
Substitute \( x = -1 \), \( y = 3 \), and \( z = 3 \) into the equation:
\[ -1 + 3 + 2(3) = \lambda \]
Simplifying this:
\[ -1 + 3 + 6 = \lambda \]
\[ 8 = \lambda \]

4. **Conclusion:**
Therefore, the value of \( \lambda \) is \( 8 \).

### Final Answer:
\[ \lambda = 8 \]