From a point `P(lambda, lambda, lambda)`, perpendicular PQ and PR are drawn respectively on the lines `y=x, z=1 and y=-x, z=-1`. If P is such tthat `angleQPR` is a right angle , then the possible value(s) of `lambda` is (are)

Options

A. (a)`sqrt(2)`
B. (b)`1`
C. (c)`-1`
D. (d)`-sqrt(2)`

Answer

To solve the problem, we need to find the possible values of \(\lambda\) such that the angle \(QPR\) is a right angle, where \(P(\lambda, \lambda, \lambda)\) is a point from which perpendiculars \(PQ\) and \(PR\) are drawn to the lines defined by \(y = x, z = 1\) and \(y = -x, z = -1\).

### Step-by-Step Solution:

1. **Identify the Lines**:
- The first line can be represented as:
\[ L_1: \frac{x}{1} = \frac{y}{1} = \frac{z - 1}{0} \]
This means that any point on this line can be expressed as \(Q(r, r, 1)\) for some parameter \(r\).
- The second line can be represented as:
\[ L_2: \frac{x}{1} = \frac{y}{-1} = \frac{z + 1}{0} \]
This means that any point on this line can be expressed as \(R(k, -k, -1)\) for some parameter \(k\).

2. **Find Vectors \(PQ\) and \(PR\)**:
- The vector \(PQ\) from point \(P(\lambda, \lambda, \lambda)\) to point \(Q(r, r, 1)\) is given by:
\[ PQ = (r - \lambda, r - \lambda, 1 - \lambda) \]
- The vector \(PR\) from point \(P(\lambda, \lambda, \lambda)\) to point \(R(k, -k, -1)\) is given by:
\[ PR = (k - \lambda, -k - \lambda, -1 - \lambda) \]

3. **Condition for Right Angle**:
- For the angle \(QPR\) to be a right angle, the dot product of vectors \(PQ\) and \(PR\) must be zero:
\[ PQ \cdot PR = 0 \]
- Calculating the dot product:
\[ (r - \lambda)(k - \lambda) + (r - \lambda)(-k - \lambda) + (1 - \lambda)(-1 - \lambda) = 0 \]
- Simplifying this expression:
\[ (r - \lambda)(k - \lambda - k - \lambda) + (1 - \lambda)(-1 - \lambda) = 0 \]
\[ (r - \lambda)(-2\lambda) + (1 - \lambda)(-1 - \lambda) = 0 \]
\[ -2\lambda(r - \lambda) - (1 - \lambda)(1 + \lambda) = 0 \]

4. **Expanding and Rearranging**:
- Expanding the second term:
\[ -2\lambda(r - \lambda) - (1 - \lambda^2) = 0 \]
- Rearranging gives:
\[ -2\lambda(r - \lambda) = 1 - \lambda^2 \]

5. **Finding Values of \(\lambda\)**:
- To find specific values for \(\lambda\), we can set \(r = \lambda\) (since \(Q\) and \(P\) coincide):
\[ -2\lambda(0) = 1 - \lambda^2 \]
\[ 1 - \lambda^2 = 0 \]
- This implies:
\[ \lambda^2 = 1 \implies \lambda = 1 \text{ or } \lambda = -1 \]

### Conclusion:
The possible values of \(\lambda\) such that the angle \(QPR\) is a right angle are:
\[ \lambda = 1 \text{ and } \lambda = -1 \]