The line `x - 6y + 11 = 0 ` bisects the join of (8, -1) and (0, k). Find the value of k.

Answer

To solve the problem, we need to find the value of \( k \) such that the line \( x - 6y + 11 = 0 \) bisects the line segment joining the points \( A(8, -1) \) and \( B(0, k) \).

### Step-by-Step Solution:

1. **Find the Midpoint of the Line Segment AB:**
The midpoint \( M \) of the line segment joining the points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, \( A(8, -1) \) and \( B(0, k) \). So,
\[ M = \left( \frac{8 + 0}{2}, \frac{-1 + k}{2} \right) = \left( 4, \frac{k - 1}{2} \right) \]

2. **Substitute the Midpoint into the Line Equation:**
The line \( x - 6y + 11 = 0 \) must pass through the midpoint \( M \). Therefore, we substitute \( x = 4 \) and \( y = \frac{k - 1}{2} \) into the equation:
\[ 4 - 6\left(\frac{k - 1}{2}\right) + 11 = 0 \]

3. **Simplify the Equation:**
Simplifying the equation:
\[ 4 - 3(k - 1) + 11 = 0 \]
\[ 4 - 3k + 3 + 11 = 0 \]
\[ 18 - 3k = 0 \]

4. **Solve for k:**
Rearranging the equation gives:
\[ 3k = 18 \]
\[ k = 6 \]

Thus, the value of \( k \) is \( 6 \).