If the lines `(x-1)/(1)=(y-3)/(1)=(z-2)/(lambda)` and `(x-1)/(lambda)=(y-3)/(2)=(z-4)/(1)` intersect at a point, then the value of `lambda^(2)+4` is equal to

Options

A. 8
B. 10
C. 13
D. 5

Answer

To solve the problem, we need to analyze the two lines given in the question and find the value of \( \lambda^2 + 4 \) when the lines intersect.

### Step 1: Write the equations of the lines in parametric form.

The first line can be expressed as:
\[ \frac{x-1}{1} = \frac{y-3}{1} = \frac{z-2}{\lambda} = \alpha \]
From this, we can derive the parametric equations:
- \( x = \alpha + 1 \)
- \( y = \alpha + 3 \)
- \( z = \lambda \alpha + 2 \)

The second line can be expressed as:
\[ \frac{x-1}{\lambda} = \frac{y-3}{2} = \frac{z-4}{1} = \beta \]
From this, we can derive the parametric equations:
- \( x = \lambda \beta + 1 \)
- \( y = 2\beta + 3 \)
- \( z = \beta + 4 \)

### Step 2: Set the parametric equations equal to each other.

Since the lines intersect, the coordinates \( (x, y, z) \) from both lines must be equal. Therefore, we can set up the following equations:
1. \( \alpha + 1 = \lambda \beta + 1 \)
2. \( \alpha + 3 = 2\beta + 3 \)
3. \( \lambda \alpha + 2 = \beta + 4 \)

### Step 3: Simplify the equations.

From the first equation:
\[ \alpha + 1 = \lambda \beta + 1 \implies \alpha = \lambda \beta \]

From the second equation:
\[ \alpha + 3 = 2\beta + 3 \implies \alpha = 2\beta \]

### Step 4: Substitute \( \alpha \) in the third equation.

Now we can substitute \( \alpha = 2\beta \) into the first equation:
\[ 2\beta = \lambda \beta \]
This gives us:
\[ \lambda \beta - 2\beta = 0 \implies \beta(\lambda - 2) = 0 \]
This means either \( \beta = 0 \) or \( \lambda = 2 \).

### Step 5: Analyze the cases.

1. **Case 1**: If \( \beta = 0 \):
- From \( \alpha = 2\beta \), we get \( \alpha = 0 \).
- Substituting \( \beta = 0 \) into the third equation:
\[ \lambda(0) + 2 = 0 + 4 \implies 2 = 4 \quad \text{(not possible)} \]

2. **Case 2**: If \( \lambda = 2 \):
- This is a valid solution.

### Step 6: Calculate \( \lambda^2 + 4 \).

Now that we have \( \lambda = 2 \):
\[ \lambda^2 + 4 = 2^2 + 4 = 4 + 4 = 8 \]

### Final Answer:
The value of \( \lambda^2 + 4 \) is \( \boxed{8} \).