To write the equation of a parabola given the directrix [tex]\( x = -9 \)[/tex] and the focus [tex]\( (-1, -1) \)[/tex], we follow these steps:
1. Identify the components:
- Focus: [tex]\((p, q) = (-1, -1)\)[/tex]
- Directrix: [tex]\( x = -9\)[/tex]
2. Understand the structure of a parabolic equation:
- For a parabola that has a vertical directrix [tex]\( x = h \)[/tex] and a focus [tex]\((p, q)\)[/tex], the general form of the equation is [tex]\((y - q)^2 = 4a(x - h)\)[/tex].
3. Calculate the distance between the focus and the directrix:
- The distance between the focus and the directrix gives us the value of [tex]\( a \)[/tex] (which is the distance from the focus to the vertex along the x-axis). This value is:
[tex]\[
a = p - h = -1 - (-9) = -1 + 9 = 8
\][/tex]
4. Substitute the values into the standard form:
- Using the standard parabolic equation [tex]\((y - q)^2 = 4a(x - h)\)[/tex]:
[tex]\[
(y - (-1))^2 = 4 \cdot 2 \cdot (x - (-9))
\][/tex]
- Simplify the expression:
[tex]\[
(y + 1)^2 = 8(x + 9)
\][/tex]
So, the equation of the parabola is:
[tex]\[
(y + 1)^2 = 8(x + 9)
\][/tex]
