Recall that a parabola is a curve where any point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix).
Let (x,y) a point on the parabola with its focus at (–5,0) and its directrix y = 2, then we can set the following equation:
[tex]\sqrt[]{(x-(-5))^2+(y-0)^2}=2-y\text{.}[/tex]
Simplifying the above result we get:
[tex]\sqrt[]{(x+5)^2+y^2}=2-y\text{.}[/tex]
Taking the above result to the power of 2 we get:
[tex]\begin{gathered} \sqrt[]{(x+5)^2+y^2}^2=(2-y)^2, \\ (x+5)^2+y^2=(2-y)^2\text{.} \end{gathered}[/tex]
Therefore:
[tex](x+5)^2+y^2=4-4y+y^2\text{.}[/tex]
Subtracting y² from the above equation we get:
[tex]\begin{gathered} (x+5)^2+y^2-y^2=4-4y+y^2\text{-}y^2, \\ (x+5)^2=4-4y^{}\text{.} \end{gathered}[/tex]
Subtracting 4 from the above equation we get:
[tex]\begin{gathered} (x+5)^2-4=4-4y-4, \\ (x+5)^2-4=-4y\text{.} \end{gathered}[/tex]
Finally, dividing the above equation by -4 we get:
[tex]\begin{gathered} \frac{(x+5)^2-4}{-4}=\frac{-4y}{-4}, \\ y=-\frac{1}{4}(x+5)^2+1. \end{gathered}[/tex]
Answer: Option B.
