Find the equation of a parabola with a focus at [tex][tex]$(-2, -3)$[/tex][/tex] and a directrix at [tex][tex]$y = -5$[/tex][/tex].

A. [tex][tex]$y = \frac{1}{4}(x - 2)^2 - 4$[/tex][/tex]
B. [tex][tex]$y = \frac{1}{4}(x + 2)^2 - 8$[/tex][/tex]
C. [tex][tex]$y = -\frac{1}{16}(x + 2)^2 - 4$[/tex][/tex]
D. [tex][tex]$y = \frac{1}{4}(x + 2)^2 - 4$[/tex][/tex]



Answer :

To find the equation of a parabola with a focus and a directrix, we can follow these steps:

1. Determine the vertex of the parabola: The vertex lies halfway between the focus and the directrix. Given the focus [tex]\((-2, -3)\)[/tex] and the directrix [tex]\(y = -5\)[/tex]:

- The midpoint or average of the y-coordinates of the focus and the directrix gives the y-coordinate of the vertex.
[tex]\[ k = \frac{-3 + (-5)}{2} = \frac{-8}{2} = -4 \][/tex]

- The x-coordinate of the vertex is the same as the x-coordinate of the focus.
[tex]\[ h = -2 \][/tex]

Thus, the vertex [tex]\((h, k)\)[/tex] is [tex]\((-2, -4)\)[/tex].

2. Calculate the distance [tex]\( p \)[/tex]: This is the distance from the vertex to either the focus or the directrix.
- The distance between [tex]\(-3\)[/tex] and [tex]\(-4\)[/tex] is:
[tex]\[ p = |-3 - (-4)| = |-3 + 4| = 1 \][/tex]

3. Form the standard equation: The standard form of the equation of a parabola that opens upwards or downwards is:
[tex]\[ y = \frac{1}{4p} (x - h)^2 + k \][/tex]

Given that [tex]\( p = 1 \)[/tex], [tex]\( h = -2 \)[/tex], and [tex]\( k = -4 \)[/tex]:
- Substitute these values into the standard form equation:
[tex]\[ y = \frac{1}{4 \cdot 1} (x - (-2))^2 + (-4) \][/tex]
Simplify:
[tex]\[ y = \frac{1}{4} (x + 2)^2 - 4 \][/tex]

Comparing this with the given options:

A) [tex]\( y = \frac{1}{4}(x-2)^2 - 4 \)[/tex]

B) [tex]\( y = \frac{1}{4}(x+2)^2 - 8 \)[/tex]

C) [tex]\( y = \frac{-1}{16}(x+2)^2 - 4 \)[/tex]

D) [tex]\( y = \frac{1}{4}(x+2)^2 - 4 \)[/tex]

The correct equation that matches our derived equation [tex]\( y = \frac{1}{4}(x + 2)^2 - 4 \)[/tex] is option D.

Therefore, the correct answer is:
[tex]\[ \boxed{4} \][/tex]