Find the equation of the parabola with its focus at [tex](-5,0)[/tex] and its directrix [tex]y=2[/tex].

A. [tex]y = -\frac{1}{4}(x+5)^2 + 1[/tex]
B. [tex]y = -\frac{1}{4}(x+1)^2 + 5[/tex]
C. [tex]y = -4(x+5)^2 + 1[/tex]
D. [tex]y = \frac{1}{4}(x+5)^2 + 1[/tex]



Answer :

To find the equation of the parabola with its focus at [tex]$(-5, 0)$[/tex] and its directrix [tex]$y=2$[/tex], let's go through the following steps:

1. Identify the Vertex of the Parabola:
The vertex of the parabola is halfway between the focus and the directrix. Since the focus's y-coordinate is 0 and the directrix is the line [tex]$y=2$[/tex], the y-coordinate of the vertex is the average of these two y-coordinates.
[tex]\[ \text{vertex}_y = \frac{0 + 2}{2} = 1 \][/tex]
The x-coordinate of the vertex is the same as that of the focus since the directrix is horizontal.
[tex]\[ \text{vertex}_x = -5 \][/tex]
Therefore, the vertex of the parabola is at [tex]$(-5, 1)$[/tex].

2. Determine the Value of 'a':
The formula for the value of 'a' in the standard form of a parabola [tex]\(y = a(x - h)^2 + k\)[/tex] is given by [tex]$a = \frac{1}{4p}$[/tex], where [tex]\(p\)[/tex] is the distance from the vertex to the focus or the directrix.

Calculate [tex]\(p\)[/tex]:
[tex]\[ p = \text{distance from vertex to focus} = |1 - 0| = 1 \][/tex]

Thus,
[tex]\[ a = \frac{1}{4p} = \frac{1}{4 \times 1} = \frac{1}{4} \][/tex]

3. Write the Equation of the Parabola:
Using the vertex [tex]\((-5, 1)\)[/tex] and the value of [tex]\(a = \frac{1}{4}\)[/tex], the equation of the parabola is:
[tex]\[ y = a(x - (-5))^2 + 1 \][/tex]
Simplify it:
[tex]\[ y = \frac{1}{4}(x + 5)^2 + 1 \][/tex]

Therefore, from the given options, the correct equation for the parabola is:
[tex]\[ \boxed{D) \ y= \frac{1}{4}(x+5)^2+1} \][/tex]