If the points A (1,3) and B (5,5) lying on a parabola are equidistant from focus, then the slope of the directrix is

Options

A. `(1)/(2)`
B. `-(1)/(2)`
C. 2
D. -2

Explanation

To solve the problem, we need to find the slope of the directrix of a parabola given that points A(1, 3) and B(5, 5) are equidistant from the focus of the parabola.

### Step-by-step Solution:

1. **Identify the Points**: We have two points A(1, 3) and B(5, 5).

2. **Calculate the Slope of the Line AB**:
- The slope (m) of the line connecting points A and B can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
- Here, let \( (x_1, y_1) = (1, 3) \) and \( (x_2, y_2) = (5, 5) \).
- Plugging in the values:
\[ m = \frac{5 - 3}{5 - 1} = \frac{2}{4} = \frac{1}{2} \]

3. **Understanding the Relationship with the Directrix**:
- Since points A and B are equidistant from the focus of the parabola, the line segment AB is perpendicular to the axis of symmetry of the parabola.
- The directrix of the parabola is parallel to this line segment AB.

4. **Finding the Slope of the Directrix**:
- If the slope of line AB is \( \frac{1}{2} \), then the slope of the directrix, which is parallel to line AB, will also be \( \frac{1}{2} \).

5. **Conclusion**:
- Therefore, the slope of the directrix is \( \frac{1}{2} \).

### Final Answer:
The slope of the directrix is \( \frac{1}{2} \).