Question

What is the vertex of a parabola that has a focus at (2,1) and the directrix at y=−3 ?

Answer 1

To find the vertex of a parabola given its focus and directrix, we can use the property that the vertex is located halfway between the focus and the directrix.

1. **Identify the focus and directrix:**
- Focus: \( (2, 1) \)
- Directrix: \( y = -3 \)

2. **Calculate the y-coordinate of the vertex:**
The y-coordinate of the vertex is halfway between the y-coordinate of the focus and the y-coordinate of the directrix.

- The y-coordinate of the focus is \( 1 \).
- The y-coordinate of the directrix is \( -3 \).

To find the midpoint, we use:
\[
y_v = \frac{y_{\text{focus}} + y_{\text{directrix}}}{2} = \frac{1 + (-3)}{2} = \frac{1 - 3}{2} = \frac{-2}{2} = -1
\]

3. **Determine the x-coordinate of the vertex:**
The vertex’s x-coordinate is the same as the focus’s x-coordinate because the parabola opens either upwards or downwards, and in this case, opens upward since the focus is above the directrix.

Therefore, the x-coordinate of the vertex is \( 2 \) (the same as the x-coordinate of the focus).

4. **Combine the results:**
The vertex \( (x_v, y_v) \) is:
\[
(2, -1)
\]

Thus, the vertex of the parabola is \( \boxed{(2, -1)} \).