If the line `(x-1)/(2)=(y+1)/(3)=(z-1)/(4) and (x-3)/(1)=(y-k)/(2)=(z)/(1)` intersect, then `k` is equal to

Options

A. a) `-1`
B. b) `(2)/(9)`
C. c) `(9)/(2)`
D. d) `0`

Answer

To solve the problem, we need to determine the value of \( k \) such that the two lines intersect. The equations of the lines are given as:

1. \(\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4}\)
2. \(\frac{x-3}{1} = \frac{y-k}{2} = \frac{z}{1}\)

### Step 1: Parameterize the equations of the lines

Let's set the first line equal to a parameter \( t \):

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

So, the first line can be expressed as:
\[ (x, y, z) = (2t + 1, 3t - 1, 4t + 1) \]

Next, let's set the second line equal to a parameter \( m \):

\[ \frac{x-3}{1} = m \implies x = m + 3 \]
\[ \frac{y-k}{2} = m \implies y = 2m + k \]
\[ \frac{z}{1} = m \implies z = m \]

So, the second line can be expressed as:
\[ (x, y, z) = (m + 3, 2m + k, m) \]

### Step 2: Set the parameterizations equal

Since the lines intersect, we can set the parameterizations equal to each other:

\[ (2t + 1, 3t - 1, 4t + 1) = (m + 3, 2m + k, m) \]

This gives us three equations:

1. \( 2t + 1 = m + 3 \)
2. \( 3t - 1 = 2m + k \)
3. \( 4t + 1 = m \)

### Step 3: Solve the equations

From the first equation:
\[ m = 2t + 1 - 3 = 2t - 2 \quad \text{(Equation 1)} \]

Substituting \( m \) from Equation 1 into the third equation:
\[ 4t + 1 = 2t - 2 \]
\[ 4t - 2t = -2 - 1 \]
\[ 2t = -3 \implies t = -\frac{3}{2} \]

Now substituting \( t = -\frac{3}{2} \) back into Equation 1 to find \( m \):
\[ m = 2\left(-\frac{3}{2}\right) - 2 = -3 - 2 = -5 \]

### Step 4: Substitute \( t \) and \( m \) into the second equation

Now we substitute \( t \) and \( m \) into the second equation:
\[ 3t - 1 = 2m + k \]
Substituting \( t = -\frac{3}{2} \) and \( m = -5 \):
\[ 3\left(-\frac{3}{2}\right) - 1 = 2(-5) + k \]
\[ -\frac{9}{2} - 1 = -10 + k \]
\[ -\frac{9}{2} - \frac{2}{2} = -10 + k \]
\[ -\frac{11}{2} = -10 + k \]
Adding 10 to both sides:
\[ k = 10 - \frac{11}{2} \]
Converting 10 to a fraction:
\[ k = \frac{20}{2} - \frac{11}{2} = \frac{9}{2} \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{\frac{9}{2}} \]