If the lines `(x-3)/(2)=(y+1)/(3)=(z-2)/(4)and(x-4)/(2)=(y-k)/(2)=(z)/(1)` intersect, then find the value of k.

Options

A. `(9)/(2)`
B. `(2)/(9)`
C. `-(3)/(2)`
D. `-(5)/(6)`

Answer

To find the value of \( k \) such that the lines

\[ \frac{x-3}{2} = \frac{y+1}{3} = \frac{z-2}{4} \]

and

\[ \frac{x-4}{2} = \frac{y-k}{2} = \frac{z}{1} \]

intersect, we will follow these steps:

### Step 1: Parametrize the lines

For the first line, let

\[ \frac{x-3}{2} = \frac{y+1}{3} = \frac{z-2}{4} = \lambda \]

From this, we can express \( x, y, z \) in terms of \( \lambda \):

- \( x = 2\lambda + 3 \)
- \( y = 3\lambda - 1 \)
- \( z = 4\lambda + 2 \)

For the second line, let

\[ \frac{x-4}{2} = \frac{y-k}{2} = \frac{z}{1} = \alpha \]

From this, we can express \( x, y, z \) in terms of \( \alpha \):

- \( x = 2\alpha + 4 \)
- \( y = 2\alpha + k \)
- \( z = \alpha \)

### Step 2: Set the coordinates equal

Since the lines intersect, the coordinates must be equal at the point of intersection:

1. \( 2\lambda + 3 = 2\alpha + 4 \) (for \( x \))
2. \( 3\lambda - 1 = 2\alpha + k \) (for \( y \))
3. \( 4\lambda + 2 = \alpha \) (for \( z \))

### Step 3: Solve for \( \alpha \) in terms of \( \lambda \)

From the third equation:

\[ \alpha = 4\lambda + 2 \]

### Step 4: Substitute \( \alpha \) into the first equation

Substituting \( \alpha \) into the first equation:

\[ 2\lambda + 3 = 2(4\lambda + 2) + 4 \]

Expanding the right side:

\[ 2\lambda + 3 = 8\lambda + 4 + 4 \]
\[ 2\lambda + 3 = 8\lambda + 8 \]

Rearranging gives:

\[ 2\lambda - 8\lambda = 8 - 3 \]
\[ -6\lambda = 5 \implies \lambda = -\frac{5}{6} \]

### Step 5: Substitute \( \lambda \) back to find \( \alpha \)

Now substituting \( \lambda \) back into the equation for \( \alpha \):

\[ \alpha = 4\left(-\frac{5}{6}\right) + 2 = -\frac{20}{6} + 2 = -\frac{20}{6} + \frac{12}{6} = -\frac{8}{6} = -\frac{4}{3} \]

### Step 6: Substitute \( \lambda \) and \( \alpha \) into the second equation

Now substitute \( \lambda \) and \( \alpha \) into the second equation:

\[ 3\left(-\frac{5}{6}\right) - 1 = 2\left(-\frac{4}{3}\right) + k \]

Calculating the left side:

\[ -\frac{15}{6} - 1 = -\frac{15}{6} - \frac{6}{6} = -\frac{21}{6} \]

Now the equation becomes:

\[ -\frac{21}{6} = -\frac{8}{3} + k \]

### Step 7: Solve for \( k \)

Convert \(-\frac{8}{3}\) to sixths:

\[ -\frac{8}{3} = -\frac{16}{6} \]

So the equation is:

\[ -\frac{21}{6} = -\frac{16}{6} + k \]

Rearranging gives:

\[ k = -\frac{21}{6} + \frac{16}{6} = -\frac{5}{6} \]

### Final Answer

Thus, the value of \( k \) is

\[ \boxed{-\frac{5}{6}} \]