The line passing through (1, 2, 3) and having direction ratios given by lt 1, 2, 3 gt cuts the x-axis distance k form origin.
What is the value of k?

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To solve the problem, we need to find the distance \( k \) from the origin to the point where the line intersects the x-axis. Let's break down the solution step by step.

### Step 1: Write the equation of the line
Given a point \( (1, 2, 3) \) and direction ratios \( \langle 1, 2, 3 \rangle \), we can write the parametric equations of the line as follows:

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

This implies:
- \( x = 1 + t \)
- \( y = 2 + 2t \)
- \( z = 3 + 3t \)

### Step 2: Determine where the line intersects the x-axis
The line intersects the x-axis when both \( y \) and \( z \) are equal to zero. Therefore, we set:

\[ y = 0 \quad \text{and} \quad z = 0 \]

Substituting \( y = 0 \) into the equation for \( y \):

\[ 2 + 2t = 0 \implies 2t = -2 \implies t = -1 \]

Now substituting \( t = -1 \) into the equation for \( z \):

\[ 3 + 3(-1) = 0 \implies 3 - 3 = 0 \]

Both conditions are satisfied with \( t = -1 \).

### Step 3: Find the corresponding \( x \) coordinate
Now, substituting \( t = -1 \) into the equation for \( x \):

\[ x = 1 + (-1) = 0 \]

Thus, the point of intersection of the line with the x-axis is \( (0, 0, 0) \).

### Step 4: Calculate the distance \( k \) from the origin
The distance \( k \) from the origin \( (0, 0, 0) \) to the point \( (0, 0, 0) \) is simply:

\[ k = \sqrt{(0 - 0)^2 + (0 - 0)^2 + (0 - 0)^2} = \sqrt{0} = 0 \]

### Final Answer
The value of \( k \) is:

\[ \boxed{0} \]