Parabola: equation y² = 4ax, parametric form (at², 2at), focus and directrix
Focus-Directrix: every point on the parabola is equidistant from focus and directrix
Rectangular Hyperbola: equation xy = c², parametric form (ct, c/t), asymptotes along axes
General points: algebraic points on a curve that move with parameter t (or p, q)
Tangents: found by differentiating y w.r.t. x at a general point, then using y − y₁ = m(x − x₁)
Normals: perpendicular to tangent, gradient = −1/m
What is a Parabola?
A parabola is a U-shaped quadratic curve with a single line of symmetry. You've seen y = x² already — that's a parabola with the y-axis as its line of symmetry.
In this chapter we work with parabolas whose line of symmetry is the x-axis — they open to the right. Think of the curve as a satellite dish lying on its side.
Parabolas belong to the family of conic sections, which also includes ellipses and hyperbolas — all of them can be obtained by slicing a cone at different angles.
Why 4a? It looks awkward, but it's chosen so that the focus ends up at the clean coordinate (a, 0). Everything slots together perfectly.
The two branches split above and below the x-axis:
- y = √(4ax) — upper branch
- y = −√(4ax) — lower branch
Converting Between Forms
The parametric equations x = at², y = 2at describe the same curve as y² = 4ax. Here's how to go from parametric to Cartesian:
Worked Example
Show that x = at², y = 2at gives the Cartesian equation y² = 4ax.
From y = 2at, make t the subject: t = y/(2a)
t = y / (2a)
Substitute into x = at²:
x = a · (y/2a)² = a · y²/4a² = y²/4a
Rearrange:
y² = 4ax ✓
Both forms are given in your Formulae Booklet, but examiners love asking you to derive one from the other — practice this elimination every time!
Focus and Directrix
Every parabola has two special geometric features: a focus (a point inside the curve) and a directrix (a vertical line outside the curve).
Example: For the parabola y² = 12x, we have 4a = 12, so a = 3. The focus is at (3, 0) and the directrix is x = −3.
How to find a: Always compare your equation to y² = 4ax. Whatever number is on the right, divide it by 4. That's a.
The Focus-Directrix Property
This is the defining geometric property of a parabola — it's the reason the curve has the shape it does.
In plain English: every point on a parabola is exactly the same distance from the focus as it is (horizontally) from the directrix. The parabola is the locus (set) of all such points.
Analogy:
Proof — Focus-Directrix → y² = 4ax
Let P = (x, y), focus S = (a, 0), directrix x = −a. Show PS = PX implies y² = 4ax.