Q.Find the equation of the parabola that satisfies the given conditions: Focus $(6, 0)$; directrix $x = -6$.

Concept: Parabola Standard Form — When the focus lies on the $x$-axis and the directrix is vertical, the parabola opens left or right. The vertex is midway between focus and directrix.

Step 1: Focus is $(6,0)$ and directrix is $x = -6$. The vertex is at the midpoint:

$x$-coordinate = $\frac{6 + (-6)}{2} = 0$, $y$-coordinate = $0$. So vertex = $(0,0)$. …

This is a classic problem where the definition of a parabola — the set of points equidistant from a fixed point (focus) and a fixed line (directrix) — gives you the equation directly. No memorising forms needed; just apply the distance condition.

The focus is $(6,0)$ and the directrix is the vertical line $x = -6$. Since the directrix is vertical, the parabola opens horizontally. The focus lies to the right of the directrix, so the parabola opens to the right.

Let’s work through it step by step.

1. Set up the distance condition. Let $P(x, y)$ be any point on the parabola. By definition, the distance from $P$ to the focus equals the perpendicular distance from $P$ to the directrix. Distance to focus $(6,0)$:

Distance to directrix $x = -6$: the perpendicular distance from a point to a vertical line is the absolute horizontal difference:

2. Equate the two distances.

3. Square both sides (both sides are non-negative, so no sign issues):

4. Expand and simplify.

Cancel $x^2$ and $36$ from both sides:

That’s the equation. …