Answer:
Option B: [tex]$ \textbf{y} \hspace{1mm} \textbf{=} \hspace{1mm} \frac{\textbf{-1}}{\textbf{2}} \textbf{x}^{\textbf{2}} \hspace{1mm} \textbf{-} \hspace{1mm} \frac{\textbf{5}}{\textbf{2}} $[/tex]
Step-by-step explanation:
When the focus (h, k) of a parabola and the equation of the directrix y = c are given, the equation of the parabola is given by:
[tex]$ \textbf{(x - h)}^{\textbf{2}} \hspace{1mm} \textbf{+} \hspace{1mm} \textbf{k}^{\textbf{2}} \hspace{1mm} \textbf{-} \hspace{1mm} \textbf{c}^{\textbf{2}} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{2(k - c)y}} $[/tex]
Here, we are given the focus: (h, k) = (0, -2)
Directrix: y = c = -3.
We substitute in the formula to get the equation of the parabola.
[tex]$ (x - 0)^2 + (-2)^2 - (-3)^2 = 2(-2 - (-3))y $[/tex]
[tex]$ \implies x^2 + 4 - 9 = 2(- 2 + 3)y $[/tex]
[tex]$ \implies x^2 - 5 = 2(1) y$[/tex]
[tex]$ \implies 2y = x^2 - 5 $[/tex]
Dividing by 2, throughout we get:
[tex]$ \textbf{y} \hspace{1mm} \textbf{=} \hspace{1mm} \frac{\textbf{-1}}{\textbf{2}} \textbf{x}^{\textbf{2}} \hspace{1mm} \textbf{-} \hspace{1mm} \frac{\textbf{5}}{\textbf{2}} $[/tex] which is the required answer.