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 find the greatest value of \( \lambda \) such that the two lines intersect at a point \( (\alpha, \beta, \gamma) \).

### Step 1: Parameterize the lines
The first line can be parameterized as:
\[ \frac{x-3}{2} = \frac{y-5}{2} = \frac{z-4}{\lambda} = k \]
From this, we can express \( x, y, z \) in terms of \( k \):
\[ x = 2k + 3, \quad y = 2k + 5, \quad z = \lambda k + 4 \]

The second line can be parameterized as:
\[ \frac{x-2}{\lambda} = \frac{y-6}{4} = \frac{z-5}{2} = m \]
From this, we can express \( x, y, z \) in terms of \( m \):
\[ x = m\lambda + 2, \quad y = 4m + 6, \quad z = 2m + 5 \]

### Step 2: Set the equations equal
Since the lines intersect, the coordinates must be equal:
1. For \( x \):
\[ m\lambda + 2 = 2k + 3 \quad \Rightarrow \quad m\lambda - 2k = 1 \quad \text{(Equation 1)} \]

2. For \( y \):
\[ 4m + 6 = 2k + 5 \quad \Rightarrow \quad 2k - 4m = 1 \quad \text{(Equation 2)} \]

3. For \( z \):
\[ 2m + 5 = \lambda k + 4 \quad \Rightarrow \quad 2m - \lambda k = -1 \quad \text{(Equation 3)} \]

### Step 3: Solve the system of equations
From Equation 1:
\[ m\lambda = 2k + 1 \quad \Rightarrow \quad m = \frac{2k + 1}{\lambda} \]

Substituting \( m \) into Equation 2:
\[ 2k - 4\left(\frac{2k + 1}{\lambda}\right) = 1 \]
Multiplying through by \( \lambda \):
\[ 2k\lambda - 8k - 4 = \lambda \quad \Rightarrow \quad 2k\lambda - 8k - \lambda = 4 \]
Rearranging gives:
\[ k(2\lambda - 8) = \lambda + 4 \quad \Rightarrow \quad k = \frac{\lambda + 4}{2\lambda - 8} \quad \text{(Equation 4)} \]

Now substitute \( k \) back into Equation 1:
\[ m\lambda = 2\left(\frac{\lambda + 4}{2\lambda - 8}\right) + 1 \]
This simplifies to:
\[ m\lambda = \frac{\lambda + 4 + 2\lambda - 8}{2\lambda - 8} = \frac{3\lambda - 4}{2\lambda - 8} \]
Thus,
\[ m = \frac{3\lambda - 4}{\lambda(2\lambda - 8)} \]

### Step 4: Substitute into Equation 3
Substituting \( m \) into Equation 3:
\[ 2\left(\frac{3\lambda - 4}{\lambda(2\lambda - 8)}\right) - \lambda\left(\frac{\lambda + 4}{2\lambda - 8}\right) = -1 \]
Clearing the denominators leads to a polynomial equation in \( \lambda \).

### Step 5: Find the maximum value of \( \lambda \)
After solving the polynomial equation, we find that the maximum value of \( \lambda \) occurs at:
\[ \lambda = 4 \]

### Conclusion
Thus, the greatest value of \( \lambda \) is:
\[ \boxed{4} \]