If the lines `(x-3)/(2)=(y-5)/(2)=(z-4)/(lambda) and (x-2)/(lambda)=(y-6)/(4)=(z-5)/(2)` intersect at a point `(alpha, beta, gamma)`, then the greatest value of `lambda` is equal to

Options

A. 0
B. 2
C. `-2`
D. 4

Answer

To solve the problem, we need to determine the greatest value of \( \lambda \) for which the two given lines intersect. The lines are represented in symmetric form, and we will use the condition for coplanarity of the two lines to find the value of \( \lambda \).

### Step-by-Step Solution:

1. **Identify the Lines:**
The first line is given by:
\[ \frac{x-3}{2} = \frac{y-5}{2} = \frac{z-4}{\lambda} \]
The second line is given by:
\[ \frac{x-2}{\lambda} = \frac{y-6}{4} = \frac{z-5}{2} \]

2. **Extracting Points and Direction Ratios:**
From the first line, we can identify:
- A point on line 1: \( (3, 5, 4) \)
- Direction ratios of line 1: \( (2, 2, \lambda) \)

From the second line, we can identify:
- A point on line 2: \( (2, 6, 5) \)
- Direction ratios of line 2: \( (\lambda, 4, 2) \)

3. **Setting Up the Determinant for Coplanarity:**
For the two lines to intersect, the following determinant must equal zero:
\[ \begin{vmatrix} x_2 - x_1 & a_1 & a_2 \\ y_2 - y_1 & b_1 & b_2 \\ z_2 - z_1 & c_1 & c_2 \end{vmatrix} = 0 \]
Substituting the values:
- \( x_2 - x_1 = 2 - 3 = -1 \)
- \( y_2 - y_1 = 6 - 5 = 1 \)
- \( z_2 - z_1 = 5 - 4 = 1 \)

The determinant becomes:
\[ \begin{vmatrix} -1 & 2 & \lambda \\ 1 & 2 & 4 \\ 1 & \lambda & 2 \end{vmatrix} = 0 \]

4. **Calculating the Determinant:**
Expanding the determinant:
\[ -1 \begin{vmatrix} 2 & 4 \\ \lambda & 2 \end{vmatrix} - 2 \begin{vmatrix} 1 & 4 \\ 1 & 2 \end{vmatrix} + \lambda \begin{vmatrix} 1 & 2 \\ 1 & 2 \end{vmatrix} \]
Calculating the 2x2 determinants:
- \( \begin{vmatrix} 2 & 4 \\ \lambda & 2 \end{vmatrix} = 4 - 2\lambda \)
- \( \begin{vmatrix} 1 & 4 \\ 1 & 2 \end{vmatrix} = 2 - 4 = -2 \)
- \( \begin{vmatrix} 1 & 2 \\ 1 & 2 \end{vmatrix} = 0 \)

Thus, the determinant simplifies to:
\[ -1(4 - 2\lambda) - 2(-2) + 0 = 0 \]
Which simplifies to:
\[ -4 + 2\lambda + 4 = 0 \implies 2\lambda = 0 \implies \lambda = 0 \]

5. **Finding the Greatest Value of \( \lambda \):**
The only value we found for \( \lambda \) is \( 0 \). Since we are asked for the greatest value, we conclude that:
\[ \text{Greatest value of } \lambda = 0 \]

### Final Answer:
The greatest value of \( \lambda \) is \( 0 \).