To find the equation of a parabola given a focus and a directrix, follow these steps:
1. Identify the Focus and Directrix:
- Focus: [tex]\( (10, -6) \)[/tex]
- Directrix: [tex]\( x = 6 \)[/tex]
2. Determine the Vertex:
The vertex of a parabola lies midway between the focus and the directrix. If the directrix is a vertical line [tex]\( x = 6 \)[/tex], then it is vertical by symmetry about an axis parallel to the y-axis.
The x-coordinate of the vertex is the midpoint between the x-coordinate of the focus and the directrix:
[tex]\[
h = \frac{10 + 6}{2} = 8.0
\][/tex]
The y-coordinate of the vertex remains the same as the y-coordinate of the focus since the directrix is a vertical line:
[tex]\[
k = -6
\][/tex]
3. Calculate the Distance [tex]\( p \)[/tex] (distance from the vertex to the focus):
[tex]\[
p = 10 - h = 10 - 8 = 2.0
\][/tex]
4. Write the Equation of the Parabola:
The standard form of the equation of a parabola with a vertical axis and vertex at [tex]\( (h, k) \)[/tex] is:
[tex]\[
(x - h)^2 = 4p(y - k)
\][/tex]
Substituting [tex]\( h = 8.0 \)[/tex], [tex]\( k = -6 \)[/tex], and [tex]\( p = 2 \)[/tex]:
[tex]\[
(x - 8.0)^2 = 4 \times 2.0 (y - (-6))
\][/tex]
Simplifying this:
[tex]\[
(x - 8.0)^2 = 8.0(y + 6)
\][/tex]
Hence, the equation of the parabola is:
[tex]\[
(x - 8.0)^2 = 8.0(y + 6)
\][/tex]
Summing up:
- Vertex: [tex]\( (8.0, -6) \)[/tex]
- Distance [tex]\( p \)[/tex]: [tex]\( 2.0 \)[/tex]
- Equation of the parabola: [tex]\( (x - 8.0)^2 = 8.0(y + 6) \)[/tex]
