Compare with y² = 4ax to find a = 3, focus at (3,0).

Find Focus and Directrix of Parabola y² = 12x

Question

Find the focus and directrix of the parabola $y^2 = 12x$ .

Solution — Step by Step

The standard form of a parabola opening right is $y^2 = 4ax$ , where $a$ is the distance from the vertex to the focus. We match our equation against this form.

Comparing $y^2 = 12x$ with $y^2 = 4ax$ :

4a = 12 \implies a = 3

So the parameter $a = 3$ .

For $y^2 = 4ax$ , the focus always sits at $(a, 0)$ . Since $a = 3$ , the focus is at (3, 0).

The directrix is the vertical line $x = -a$ . With $a = 3$ :

\text{Directrix: } x = -3

The vertex is at the origin $(0, 0)$ , and the axis of symmetry is the x-axis ( $y = 0$ ). This fully characterises the parabola.

Why This Works

Every parabola is defined by a focus-directrix property: any point $P$ on the parabola is equidistant from the focus $F$ and the directrix line. The equation $y^2 = 4ax$ is the algebraic expression of exactly this geometric condition, derived by setting $PF = PM$ (where $M$ is the foot of perpendicular from $P$ to the directrix) and squaring both sides.

The parameter $a$ controls how “wide” the parabola opens. Larger $a$ means the focus is farther from the vertex, and the curve is broader. Here $a = 3$ puts the focus 3 units to the right of the vertex — so any point on $y^2 = 12x$ is exactly 3 units from the line $x = -3$ as well as from the point $(3, 0)$ .

This is why the comparison step is so powerful: once we know $4a$ , we know everything about the parabola’s shape and position.

Alternative Method (Using Latus Rectum)

We can verify by computing the latus rectum — the chord through the focus perpendicular to the axis.

For $y^2 = 4ax$ , the length of the latus rectum is $4a$ . Substituting $x = a = 3$ into the parabola:

y^2 = 12(3) = 36 \implies y = \pm 6

The latus rectum runs from $(3, -6)$ to $(3, 6)$ , length $= 12 = 4a$ . This confirms $a = 3$ and that $(3, 0)$ is indeed the focus — it lies exactly on the latus rectum’s midpoint.

In CBSE boards and JEE Main, questions often give the latus rectum length and ask for the equation of the parabola. Work backwards: if latus rectum $= 12$ , then $4a = 12$ , so $a = 3$ , and the equation is $y^2 = 12x$ . This shortcut saves time in MCQs.

Common Mistake

The most common error is writing the focus as $(12, 0)$ instead of $(3, 0)$ — students read off the coefficient directly instead of dividing by 4. Remember: the coefficient in $y^2 = 12x$ is $4a$ , not $a$ . Always divide by 4 first. Similarly, the directrix becomes $x = -12$ instead of the correct $x = -3$ . In a CBSE board exam, this costs you full marks even though the method was right.

Question

Solution — Step by Step

Why This Works

Alternative Method (Using Latus Rectum)

Common Mistake

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