Find the equation for the parabola with the following focus and directrix:

Focus: [tex]\((-2,1)\)[/tex]

Directrix: [tex]\(x=-8\)[/tex]

[tex]\[
(y - \square)^2 = \square(x + \square)
\][/tex]

Enter the correct values in the squares.



Answer :

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]