Q.Find the equation of the parabola that satisfies the given conditions: Vertex $(0, 0)$; focus $(3, 0)$.

Concept: Parabola Standard Form

Since the vertex is at $(0,0)$ and the focus is at $(3,0)$, the axis is the positive $x$-axis. The standard equation for such a parabola is $y^2 = 4ax$, where $a$ is the distance from the vertex to the focus.

Step 1: Identify $a$. …

A parabola is defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). When the vertex is at $(0,0)$ and the focus lies on the x‑axis at $(3,0)$, the axis of the parabola is the x‑axis itself. The vertex sits exactly midway between the focus and the directrix, so the directrix must be the vertical line $x = -3$.

Because the focus is to the right of the vertex, the parabola opens to the right. For a parabola with vertex at the origin and axis along the x‑axis, the standard equation is:

Here $a$ is the signed distance from the vertex to the focus. Since the focus is at $(3,0)$, we have $a = 3$. The directrix is $x = -a = -3$.

Now let’s build the equation step by step.

1. Identify the orientation.

   The focus $(3,0)$ is on the positive x‑axis and the vertex is at the origin. This tells us the parabola opens to the right. If it opened to the left, the focus would have a negative x‑coordinate.

2. Recall the standard form for a right‑opening parabola.

   When the vertex is at $(0,0)$ and the axis is horizontal, the equation is $y^2 = 4ax$. The parameter $a$ is the distance from the vertex to the focus (and also from the vertex to the directrix, but on the opposite side).

3. Find $a$ from the focus.

   The focus is $(a, 0)$. We are given $(3,0)$, so $a = 3$.

4. Write the equation.

   Substitute $a = 3$ into $y^2 = 4ax$: …