The lines `(x-1)/(2)=(y+1)/(2)=(z-1)/(4) and (x-3)/(1)=(y-6)/(2)=(z)/(1)` intersect each other at point

Options

A. (-2,-4,5)
B. (-2,-4,-5)
C. (2,4,-5)
D. (2,-4,-5)

Explanation

To find the intersection point of the given lines, we will first express both lines in parametric form and then solve for the parameters to find the intersection point.

### Step 1: Write the equations in parametric form

The first line is given by:
\[ \frac{x-1}{2} = \frac{y+1}{2} = \frac{z-1}{4} = \lambda \]
From this, we can express \(x\), \(y\), and \(z\) in terms of \(\lambda\):
- \(x = 2\lambda + 1\)
- \(y = 2\lambda - 1\)
- \(z = 4\lambda + 1\)

The second line is given by:
\[ \frac{x-3}{1} = \frac{y-6}{2} = \frac{z}{1} = \mu \]
From this, we can express \(x\), \(y\), and \(z\) in terms of \(\mu\):
- \(x = \mu + 3\)
- \(y = 2\mu + 6\)
- \(z = \mu\)

### Step 2: Set the equations equal to find the intersection

Now we will set the equations for \(x\), \(y\), and \(z\) from both lines equal to each other:

1. For \(x\):
\[ 2\lambda + 1 = \mu + 3 \quad \text{(1)} \]

2. For \(y\):
\[ 2\lambda - 1 = 2\mu + 6 \quad \text{(2)} \]

3. For \(z\):
\[ 4\lambda + 1 = \mu \quad \text{(3)} \]

### Step 3: Solve the equations

From equation (3):
\[ \mu = 4\lambda + 1 \]

Substituting \(\mu\) into equation (1):
\[ 2\lambda + 1 = (4\lambda + 1) + 3 \]
\[ 2\lambda + 1 = 4\lambda + 4 \]
Rearranging gives:
\[ 2\lambda - 4\lambda = 4 - 1 \]
\[ -2\lambda = 3 \quad \Rightarrow \quad \lambda = -\frac{3}{2} \]

Now substitute \(\lambda = -\frac{3}{2}\) back into equation (3) to find \(\mu\):
\[ \mu = 4\left(-\frac{3}{2}\right) + 1 = -6 + 1 = -5 \]

### Step 4: Find the coordinates of the intersection point

Now we can find the coordinates of the intersection point using either line. We'll use the first line:

- For \(x\):
\[ x = 2\left(-\frac{3}{2}\right) + 1 = -3 + 1 = -2 \]

- For \(y\):
\[ y = 2\left(-\frac{3}{2}\right) - 1 = -3 - 1 = -4 \]

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

Thus, the intersection point is:
\[ (-2, -4, -5) \]

### Final Answer:
The lines intersect at the point \((-2, -4, -5)\).