If the planes `barr.(2i - lambdaj + k)=3` and `barr.(4i + j - muk)=5` are parallel, then the values of `lambda` and `mu` are respectively

Options

A. `1/2,-2`
B. `-1/2,2`
C. `-1/2,-2`
D. `1/2,2`

Answer

To solve the problem, we need to find the values of \( \lambda \) and \( \mu \) such that the given planes are parallel. The equations of the planes are:

1. \( \mathbf{r} \cdot (2\mathbf{i} - \lambda \mathbf{j} + \mathbf{k}) = 3 \)
2. \( \mathbf{r} \cdot (4\mathbf{i} + \mathbf{j} - \mu \mathbf{k}) = 5 \)

### Step 1: Identify the direction ratios of the planes

The direction ratios of the first plane are given by the coefficients of \( \mathbf{i}, \mathbf{j}, \mathbf{k} \):
- For the first plane: \( (2, -\lambda, 1) \)

The direction ratios of the second plane are:
- For the second plane: \( (4, 1, -\mu) \)

### Step 2: Set up the condition for parallel planes

For two planes to be parallel, their direction ratios must be proportional. This means we can set up the following relationship:
\[ \frac{2}{4} = \frac{-\lambda}{1} = \frac{1}{-\mu} \]

### Step 3: Solve for \( \lambda \)

From the first part of the proportion:
\[ \frac{2}{4} = \frac{1}{2} \]
Thus, we have:
\[ \frac{2}{4} = \frac{-\lambda}{1} \]
This implies:
\[ \frac{1}{2} = -\lambda \]
Solving for \( \lambda \):
\[ \lambda = -\frac{1}{2} \]

### Step 4: Solve for \( \mu \)

Now, using the second part of the proportion:
\[ \frac{1}{-\mu} = \frac{1}{2} \]
This implies:
\[ -\mu = 2 \]
Solving for \( \mu \):
\[ \mu = -2 \]

### Final Answer

Thus, the values of \( \lambda \) and \( \mu \) are:
\[ \lambda = -\frac{1}{2}, \quad \mu = -2 \]