Q.Find the equation of the parabola that satisfies the given conditions: Vertex $(0, 0)$; focus $(-2, 0)$.

Concept: Parabola Standard Form — When the vertex is at the origin and the focus lies on the x-axis, the parabola opens left or right. Here the focus $(-2, 0)$ is to the left of the vertex, so the parabola opens left.

The standard form for a left-opening parabola with vertex $(0,0)$ is

where $a$ is the distance from the vertex to the focus. …

Concept and Intuition

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). When the vertex is at $(0,0)$, the parabola’s orientation is determined entirely by where the focus lies relative to the vertex.

Here, the focus is $(-2, 0)$ — two units to the left of the vertex. That tells us two things immediately:

• The parabola opens leftward (toward the focus).
• The axis of symmetry is the x‑axis (since both vertex and focus have $y = 0$).

For a left‑opening parabola with vertex at the origin, the standard equation is $y^2 = -4ax$, where $a$ is the distance from the vertex to the focus (and also from the vertex to the directrix). The negative sign makes the parabola open left instead of right.

Step‑by‑Step Solution

1. Identify the distance $a$ from vertex to focus.

The vertex is $(0,0)$ and the focus is $(-2,0)$. The distance between them is simply the absolute value of the x‑coordinate difference:

2. Determine the orientation.

Since the focus lies on the negative x‑axis (to the left of the vertex), the parabola opens left. This means the $y^2$ term is present, and the $x$ term has a negative coefficient. …