To find the equation of the parabola with the given focus and directrix, we can follow these steps:
1. Identify Important Information:
- The focus of the parabola is [tex]\((-2, 1)\)[/tex].
- The directrix of the parabola is [tex]\(x = -8\)[/tex].
2. Determine the Vertex:
The vertex [tex]\((h, k)\)[/tex] of the parabola lies halfway between the focus and the directrix.
- For the x-coordinate [tex]\(h\)[/tex], we average the x-coordinate of the focus and the x-coordinate of the directrix:
[tex]\[
h = \frac{-2 + (-8)}{2} = \frac{-10}{2} = -5
\][/tex]
- The y-coordinate [tex]\(k\)[/tex] is the same as the y-coordinate of the focus, which is [tex]\(1\)[/tex].
3. Calculate the Value of [tex]\(p\)[/tex]:
The distance [tex]\(p\)[/tex] between the vertex and the focus (or the vertex and the directrix) is the horizontal distance:
[tex]\[
p = \text{distance between } -5 \text{ and } -2 = |-2 - (-5)| = |-2 + 5| = 3
\][/tex]
4. Determine the Standard Form of the Parabola:
Given the horizontal orientation of the parabola, the standard form is:
[tex]\[
(y - k)^2 = 4p(x - h)
\][/tex]
Plugging in the values, we get:
[tex]\[
(y - 1)^2 = 4 \cdot 3 (x + 5)
\][/tex]
5. Simplify the Coefficients:
Multiplying out the right side constants:
[tex]\[
4 \cdot 3 = 12
\][/tex]
So, the equation of the parabola is:
[tex]\[
(y - 1)^2 = 12(x + 5)
\][/tex]
Therefore, the values to fill in the equation [tex]\((y-1)^2 = [?](x+\square)\)[/tex] are:
[tex]\[
(y - 1)^2 = 12(x + 5)
\][/tex]
Hence, the coefficients are:
- The coefficient of the right side term is [tex]\(12\)[/tex].
- The value that completes the binomial for [tex]\(x\)[/tex] is [tex]\(+5\)[/tex].
So,
- The equation takes the form [tex]\( (y - 1)^2 = 12(x + 5) \)[/tex].
Thus, the completed equation of the parabola is:
[tex]\[
(y-1)^2 = 12(x+5).
\][/tex]
