To find the equation of the parabola given the focus and directrix, follow these steps:
1. Identify known parts of the parabola:
- Focus: [tex]\( (-2, 1) \)[/tex]
- Directrix: [tex]\( x = -8 \)[/tex]
2. Determine the key components:
- The vertex form of a parabola with a vertical directrix is: [tex]\( (y - k)^2 = 4p(x - h) \)[/tex]
- Here, [tex]\((h, k)\)[/tex] represents the focus of the parabola.
- Given the directrix equation [tex]\( x = -8 \)[/tex], the relationship with the focus is given by [tex]\( x = h - p \)[/tex].
3. Substitute the known value of the focus:
- From the given focus, [tex]\( h = -2 \)[/tex] and [tex]\( k = 1 \)[/tex].
4. Find the value of [tex]\( p \)[/tex]:
- The directrix [tex]\( x = -8 \)[/tex] implies that:
[tex]\( -8 = -2 - p \)[/tex]
- Solving for [tex]\( p \)[/tex]:
[tex]\[ -8 = -2 - p \][/tex]
[tex]\[ p = -6 \][/tex]
5. Write the equation of the parabola:
- Substitute [tex]\( h = -2 \)[/tex], [tex]\( k = 1 \)[/tex], and [tex]\( p = -6 \)[/tex] into the vertex form equation:
[tex]\[
(y - 1)^2 = 4p(x - (-2))
\][/tex]
6. Simplify the equation:
- Substituting [tex]\( p = -6 \)[/tex] gives:
[tex]\[
(y - 1)^2 = 4(-6)(x - (-2))
\][/tex]
[tex]\[
(y - 1)^2 = -24(x + 2)
\][/tex]
So, the equation of the parabola is: [tex]\((y - 1)^2 = -24(x + 2)\)[/tex]
