To determine the equation of a parabola with a focus and a directrix, follow these steps systematically:
1. Identify the given information:
- Focus: [tex]\( (4, -1) \)[/tex]
- Directrix: [tex]\( y = -5 \)[/tex]
2. Determine the vertex of the parabola:
- The vertex lies directly between the focus and the directrix along the y-axis.
- The y-coordinate of the vertex, [tex]\( y_{\text{vertex}} \)[/tex], is the average of the y-coordinate of the focus and the directrix:
[tex]\[
y_{\text{vertex}} = \frac{-1 + (-5)}{2} = \frac{-6}{2} = -3
\][/tex]
- Since the x-coordinate of the vertex is the same as the x-coordinate of the focus (since the focus lies directly above or below), the vertex is [tex]\( (4, -3) \)[/tex].
3. Determine the distance [tex]\( p \)[/tex]:
- The distance [tex]\( p \)[/tex] is the distance from the vertex to the focus or the vertex to the directrix (they are the same distance).
- [tex]\( p = |-1 - (-3)| = |-1 + 3| = 2 \)[/tex]
4. Formulate the equation of the parabola:
- The general form of the equation for a parabola that opens vertically is:
[tex]\[
(x - h)^2 = 4p(y - k)
\][/tex]
where [tex]\( (h, k) \)[/tex] is the vertex and [tex]\( p \)[/tex] is the distance calculated above.
- Substitute [tex]\( h = 4 \)[/tex], [tex]\( k = -3 \)[/tex], and [tex]\( p = 2 \)[/tex] into the equation:
[tex]\[
(x - 4)^2 = 4 \cdot 2 \cdot (y + 3)
\][/tex]
[tex]\[
(x - 4)^2 = 8(y + 3)
\][/tex]
5. Convert to standard form (optional):
- Expand the equation to get it in standard form:
[tex]\[(x - 4)^2 = 8(y + 3)\][/tex]
[tex]\[
(x - 4)^2 - 8(y + 3) = 0
\][/tex]
[tex]\[
(x - 4)^2 - 8y - 24 = 0
\][/tex]
The final equation of the parabola is:
[tex]\[
(x - 4)^2 = 8(y + 3)
\][/tex]
Or in standard form:
[tex]\[
(x - 4)^2 - 8y - 24 = 0
\][/tex]
Alternatively:
[tex]\[
x^2 - 8x - 8y - 8 = 0
\][/tex]
These equations characterize the parabola with the given focus and directrix.
