Let ` ( lamda , 2,1)` be a point on the plane which passes through the point `( 4,-2,2)` . If the plane is perpendicular to the line joining the points `(-2,-21,29)` and `(-1,-16 ,23)` , then `((lamda)/(11))^2 - (4 lamda)/(11) -4 ` is equal to

Answer

To solve the problem, we need to find the value of the expression \(\left(\frac{\lambda}{11}\right)^2 - \frac{4\lambda}{11} - 4\) given the conditions of the points and the plane.

### Step-by-Step Solution:

1. **Identify the Points**:
We have the points \(P_1 = (-2, -21, 29)\) and \(P_2 = (-1, -16, 23)\). We also have the point \(P_3 = (4, -2, 2)\) and the point \(P_4 = (\lambda, 2, 1)\).

2. **Find the Direction Ratios of Line \(P_1P_2\)**:
The direction ratios of the line joining \(P_1\) and \(P_2\) can be calculated as follows:
\[ \text{Direction Ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) = (-1 - (-2), -16 - (-21), 23 - 29) = (1, 5, -6) \]

3. **Find the Direction Ratios of Line \(P_3P_4\)**:
The direction ratios of the line joining \(P_3\) and \(P_4\) are:
\[ \text{Direction Ratios} = (\lambda - 4, 2 - (-2), 1 - 2) = (\lambda - 4, 4, -1) \]

4. **Condition for Perpendicularity**:
Since the plane is perpendicular to the line \(P_1P_2\), the dot product of the direction ratios must equal zero:
\[ (\lambda - 4, 4, -1) \cdot (1, 5, -6) = 0 \]
Expanding this gives:
\[ (\lambda - 4) \cdot 1 + 4 \cdot 5 + (-1) \cdot (-6) = 0 \]
Simplifying:
\[ \lambda - 4 + 20 + 6 = 0 \implies \lambda + 22 = 0 \implies \lambda = -22 \]

5. **Substitute \(\lambda\) into the Expression**:
Now we substitute \(\lambda = -22\) into the expression:
\[ \left(\frac{-22}{11}\right)^2 - \frac{4(-22)}{11} - 4 \]
Simplifying each term:
\[ \left(-2\right)^2 - \frac{-88}{11} - 4 = 4 + 8 - 4 \]

6. **Final Calculation**:
Now, we calculate:
\[ 4 + 8 - 4 = 8 \]

### Conclusion:
The value of the expression is \(8\).