6 In the xy-plane, the line determined by the points (2,k) and (32,k) passes through the origin. Which of the following could be the value of k ?

Options

A. 0
B. 4
C. 8
D. 16

Answer

To solve the problem, we need to find the value of \( k \) such that the line determined by the points \( (2, k) \) and \( (32, k) \) passes through the origin.

### Step-by-Step Solution:

1. **Understanding the Points**:
The points given are \( (2, k) \) and \( (32, k) \). Both points have the same y-coordinate, which is \( k \).

2. **Finding the Slope**:
The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For our points \( (2, k) \) and \( (32, k) \):
\[ m = \frac{k - k}{32 - 2} = \frac{0}{30} = 0 \]
This means the line is horizontal.

3. **Equation of the Line**:
Since the line is horizontal and passes through the origin, the equation of the line can be expressed as:
\[ y = k \]
However, since it passes through the origin, we also have \( y = 0 \) at \( x = 0 \).

4. **Setting the Line Through the Origin**:
For the line to pass through the origin, the y-coordinate \( k \) must be equal to 0. Thus, we have:
\[ k = 0 \]

5. **Conclusion**:
The only value of \( k \) that satisfies the condition that the line passes through the origin is:
\[ k = 0 \]

### Final Answer:
The value of \( k \) could be \( 0 \).