If `p _(1) =0 and p _(2) =0` be two non-parallel planes, then the equation `p_(1) + lamda p _(2) =0, lamda in R` represents the family of all planes through the line of intersection of the planes `p _(1) =0 and p _(2) =0` except the plane :

Options

A. `p _(1) =0`
B. `p _(2) =0`
C. `p _(1) + p _(2) =0`
D. `p _(1) -p_(2) =0`

Answer

To solve the problem, we need to analyze the given conditions about the two planes \( p_1 = 0 \) and \( p_2 = 0 \), and how they relate to the family of planes represented by the equation \( p_1 + \lambda p_2 = 0 \), where \( \lambda \) is a real number.

### Step-by-Step Solution:

1. **Understanding the Planes**:
- We have two non-parallel planes defined by the equations \( p_1 = 0 \) and \( p_2 = 0 \). The intersection of these two planes will form a line in three-dimensional space.

2. **Equation of the Family of Planes**:
- The equation \( p_1 + \lambda p_2 = 0 \) represents a family of planes that pass through the line of intersection of the two given planes. Here, \( \lambda \) can take any real value.

3. **Finding the Condition for the Family of Planes**:
- For the equation \( p_1 + \lambda p_2 = 0 \) to represent a valid plane, \( p_2 \) must not equal zero. If \( p_2 = 0 \), the equation simplifies to \( p_1 = 0 \), which corresponds to the first plane and does not represent a new plane.

4. **Identifying the Excluded Plane**:
- Since \( p_2 \) cannot be zero for the equation to represent a family of planes, the only plane that is excluded from this family is the plane defined by \( p_2 = 0 \).

5. **Conclusion**:
- Therefore, the equation \( p_1 + \lambda p_2 = 0 \) represents all planes through the line of intersection of the planes \( p_1 = 0 \) and \( p_2 = 0 \), except the plane \( p_2 = 0 \).

### Final Answer:
The plane that is excluded from the family of planes represented by \( p_1 + \lambda p_2 = 0 \) is \( p_2 = 0 \).