Question

The focus of a parabola is (0, - 2) The directrix of the parabola is the line y = - 3 What is the equation of the parabola

Answer 1

Answer:

Option B: $\textbf{y} \hspace{1mm} \textbf{=} \hspace{1mm} \frac{\textbf{-1}}{\textbf{2}} \textbf{x}^{\textbf{2}} \hspace{1mm} \textbf{-} \hspace{1mm} \frac{\textbf{5}}{\textbf{2}}$

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:

                     $\textbf{(x - h)}^{\textbf{2}} \hspace{1mm} \textbf{+} \hspace{1mm} \textbf{k}^{\textbf{2}} \hspace{1mm} \textbf{-} \hspace{1mm} \textbf{c}^{\textbf{2}} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{2(k - c)y}}$

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.

$(x - 0)^2 + (-2)^2 - (-3)^2 = 2(-2 - (-3))y$

$\implies x^2 + 4 - 9 = 2(- 2 + 3)y$

$\implies x^2 - 5 = 2(1) y$

$\implies 2y = x^2 - 5$

Dividing by 2, throughout we get:

$\textbf{y} \hspace{1mm} \textbf{=} \hspace{1mm} \frac{\textbf{-1}}{\textbf{2}} \textbf{x}^{\textbf{2}} \hspace{1mm} \textbf{-} \hspace{1mm} \frac{\textbf{5}}{\textbf{2}}$ which is the required answer.