Q.If the focus of a parabola is $(0, -3)$ and its directrix is $y = 3$, then its equation is
(A) $x^2 = -12y$
(B) $x^2 = 12y$
(C) $y^2 = -12x$
(D) $y^2 = 12x$

The key idea is to identify the type of parabola and its parameters from the given focus and directrix to use the appropriate standard form.

The focus is $(0, -3)$ and the directrix is $y = 3$.

1. Since the directrix is a horizontal line ($y=k$) and the focus is on the y-axis, the parabola opens along the y-axis.
2. The vertex of the parabola is the midpoint of the segment connecting the focus to the directrix along the axis of symmetry. Here, the vertex is $(\frac{0+0}{2}, \frac{-3+3}{2}) = (0, 0)$. …

A parabola is a fundamental conic section defined by a unique geometric property: every point on the parabola is equidistant from a specific fixed point, called the focus, and a specific fixed line, called the directrix. This definition is the cornerstone for deriving the equation of any parabola.

When we are given the focus and directrix, our goal is to use this definition to find the algebraic relationship between the $x$ and $y$ coordinates of any point on the parabola. Alternatively, we can identify the type of parabola (its orientation and vertex) and then use its standard form.

Let's break down the problem using the standard form approach, which is often quicker once the parameters are identified.

1. Identify the Focus and Directrix:

   We are given the focus $F = (0, -3)$ and the directrix $y = 3$.

2. Determine the Axis of Symmetry and Orientation:

   The axis of symmetry of a parabola is the line that passes through the focus and is perpendicular to the directrix.

   • The directrix is the horizontal line $y = 3$.
   • The focus is $(0, -3)$.
   • A line perpendicular to a horizontal line ($y=k$) is a vertical line ($x=c$).
   • Since the axis of symmetry must pass through the focus $(0, -3)$, its equation is $x = 0$ (the y-axis).
   • Now, consider the relative positions of the focus and directrix. The focus $(0, -3)$ is below the directrix $y = 3$. This means the parabola must open downwards.

3. Find the Vertex:

   The vertex of a parabola is the midpoint of the perpendicular segment from the focus to the directrix. It also lies on the axis of symmetry.

   • The focus is $(0, -3)$.
   • The point on the directrix $y=3$ that is closest to the focus (and forms the perpendicular segment) is $(0, 3)$.
   • The midpoint of $(0, -3)$ and $(0, 3)$ is $\left(\frac{0+0}{2}, \frac{-3+3}{2}\right) = (0, 0)$.
   • So, the vertex of the parabola is at the origin, $(0, 0)$.

4. Determine the Value of 'a':

   For a parabola with its vertex at the origin, the distance from the vertex to the focus (and also from the vertex to the directrix) is denoted by $|a|$. …