Q.If the points $(0, 4)$ and $(0, 2)$ are respectively the vertex and focus of a parabola, then find the equation of the parabola.

Concept: Parabola in standard form — vertex at $(h,k)$ and focus at $(h,k+a)$ gives $a$ and the opening direction.

Step 1: Vertex is $(0,4)$ and focus is $(0,2)$. Since the $x$-coordinates are the same, the axis is vertical. The focus is below the vertex, so the parabola opens downward.

Step 2: For a vertical parabola, the distance from vertex to focus is $|a|$. Here

$a = 2 - 4 = -2$ (negative confirms downward opening). …

1. Understanding the standard form

A parabola is defined as the set of points equidistant from a fixed point (focus) and a fixed line (directrix). When the axis is vertical, the standard form with vertex at $(h,k)$ is:

Here $a$ is the signed distance from the vertex to the focus.

• If $a > 0$, the parabola opens upward (focus above vertex).
• If $a < 0$, it opens downward (focus below vertex).

The focus lies on the axis, at $(h, k+a)$. The directrix is the horizontal line $y = k - a$.

2. Identifying the given points

Vertex: $(0,4)$ → so $h = 0$, $k = 4$.

Focus: $(0,2)$.

Since both have the same $x$-coordinate ($0$), the axis is the vertical line $x = 0$. The focus is below the vertex (2 < 4), so the parabola opens downward.

3. Finding $a$

The distance from vertex to focus is $|k+a - k| = |a|$.

Here $a = 2 - 4 = -2$.

So $a = -2$ (negative confirms downward opening).

4. Writing the equation

Plug $h=0$, $k=4$, $a=-2$ into $(x-h)^2 = 4a(y-k)$: …