If `P` is any point on the plane `l x+m y+n z=pa n dQ` is a point on the line `O P` such that `O PdotO Q=p^2` , then find the locus of the point `Qdot`

Explanation

To solve the problem, we need to find the locus of the point \( Q \) given that \( P \) is a point on the plane defined by the equation \( l x + m y + n z = p \) and \( Q \) is a point on the line \( OP \) such that \( OP \cdot OQ = p^2 \).

### Step-by-Step Solution:

1. **Define Points**:
Let \( P \) be represented by the coordinates \( (x_P, y_P, z_P) \) and let \( Q \) be represented by the coordinates \( (x_Q, y_Q, z_Q) \).

2. **Direction Ratios**:
The direction ratios of the line segment \( OP \) can be represented as \( (x_P, y_P, z_P) \). Since \( Q \) lies on the line \( OP \), we can express \( Q \) in terms of a parameter \( k \):
\[ Q = kP = (kx_P, ky_P, kz_P) \]

3. **Collinearity Condition**:
Since \( OP \) and \( OQ \) are collinear, we can write:
\[ \frac{x_P}{x_Q} = \frac{y_P}{y_Q} = \frac{z_P}{z_Q} = k \]

4. **Distance Condition**:
The given condition states that \( OP \cdot OQ = p^2 \). The distances can be expressed as:
\[ OP = \sqrt{x_P^2 + y_P^2 + z_P^2} \]
\[ OQ = \sqrt{x_Q^2 + y_Q^2 + z_Q^2} = \sqrt{(kx_P)^2 + (ky_P)^2 + (kz_P)^2} = k \sqrt{x_P^2 + y_P^2 + z_P^2} \]

5. **Set Up the Equation**:
Using the distance condition:
\[ OP \cdot OQ = \sqrt{x_P^2 + y_P^2 + z_P^2} \cdot k \sqrt{x_P^2 + y_P^2 + z_P^2} = p^2 \]
Simplifying gives:
\[ k (x_P^2 + y_P^2 + z_P^2) = p^2 \]
Thus,
\[ k = \frac{p^2}{x_P^2 + y_P^2 + z_P^2} \]

6. **Substituting Back**:
Substitute \( k \) back into the coordinates of \( Q \):
\[ Q = \left( \frac{p^2 x_P}{x_P^2 + y_P^2 + z_P^2}, \frac{p^2 y_P}{x_P^2 + y_P^2 + z_P^2}, \frac{p^2 z_P}{x_P^2 + y_P^2 + z_P^2} \right) \]

7. **Finding the Locus**:
Since \( P \) lies on the plane \( l x + m y + n z = p \), we can substitute \( x_P, y_P, z_P \) into the equation of the locus:
\[ l \left( \frac{p^2 x_P}{x_P^2 + y_P^2 + z_P^2} \right) + m \left( \frac{p^2 y_P}{x_P^2 + y_P^2 + z_P^2} \right) + n \left( \frac{p^2 z_P}{x_P^2 + y_P^2 + z_P^2} \right) = x_Q^2 + y_Q^2 + z_Q^2 \]

8. **Final Equation**:
After simplification, we find that the locus of point \( Q \) is given by:
\[ l x + m y + n z = x^2 + y^2 + z^2 \]

### Final Answer:
The locus of the point \( Q \) is given by the equation:
\[ l x + m y + n z = x^2 + y^2 + z^2 \]