Q.Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum of the parabola $y^2 = 12x$.

Concept: Parabola Standard Form

The given equation $y^2 = 12x$ matches the standard form $y^2 = 4ax$, where the axis is the $x$-axis, the vertex is at the origin, and $4a = 12$.

Step 1: Find $a$

$4a = 12 \implies a = 3$.

Step 2: Focus

For $y^2 = 4ax$, the focus is at $(a, 0)$.

So focus = $(3, 0)$.

Step 3: Axis

The axis is the $x$-axis (the line $y = 0$).

Step 4: Directrix and latus rectum

Directrix: $x = -a \implies x = -3$.

Length of latus rectum: $4a = 12$.

The equation $y^2 = 12x$ is already in its simplest form. When you see a parabola written as $y^2 = 4ax$, it tells you a very specific story: the vertex is at the origin $(0,0)$, the parabola opens to the right (since the $y$ term is squared and the coefficient of $x$ is positive), and the number $a$ is the distance from the vertex to the focus, as well as from the vertex to the directrix.

Why does this matter? Because once you know $a$, you can immediately write down all the key features — focus, directrix, axis, and latus rectum — without any heavy algebra. The whole problem is just about identifying $a$ correctly.

Let’s go step by step.

1. Identify $a$ from the standard form

   The given equation is $y^2 = 12x$. Compare it with $y^2 = 4ax$.

   This gives $4a = 12$, so $a = 3$.

   That’s the only number you need.

2. Focus

   For $y^2 = 4ax$, the focus lies on the axis of symmetry at $(a, 0)$.

   So here, the focus is at $(3, 0)$.

3. Axis of the parabola

   The axis is the line that passes through the vertex and the focus. Since the vertex is at $(0,0)$ and the focus is at $(3,0)$, the axis is the x-axis.

   Its equation is $y = 0$.

4. Directrix

   The directrix is a vertical line on the opposite side of the vertex from the focus, at a distance $a$ from the vertex.

   For $y^2 = 4ax$, the directrix is $x = -a$.

   So here, $x = -3$.

5. Length of the latus rectum

   The latus rectum is the chord through the focus perpendicular to the axis. Its length is always $|4a|$ for a parabola in this form.

   Since $a = 3$, the length is $4 \times 3 = 12$.