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Directrix of a Conic Section — Definition, Formula & Examples

The directrix of a conic section is a fixed line used to define the curve. Every point on the conic has a specific ratio between its distance to the focus and its distance to the directrix, and that ratio is the eccentricity.

A directrix is a fixed line ℓ\ell such that for every point PP on the conic, the ratio of the distance from PP to a focus FF to the perpendicular distance from PP to ℓ\ell equals the eccentricity ee of the conic: PFPd=e\dfrac{PF}{Pd} = e, where e<1e < 1 for an ellipse, e=1e = 1 for a parabola, and e>1e > 1 for a hyperbola.

Key Formula

PFPd=e\frac{PF}{Pd} = e
Where:
  • PFPF = Distance from a point P on the conic to the focus F
  • PdPd = Perpendicular distance from point P to the directrix
  • ee = Eccentricity of the conic (e = 1 for parabola, e < 1 for ellipse, e > 1 for hyperbola)

How It Works

The directrix works together with the focus to generate every conic section through the focus-directrix property. For a parabola y=14px2y = \frac{1}{4p}x^2 with vertex at the origin, the focus sits at (0,p)(0, p) and the directrix is the horizontal line y=−py = -p. Every point on the parabola is equidistant from the focus and the directrix. Ellipses and hyperbolas each have two directrices, one associated with each focus. The closer the eccentricity is to zero, the farther the directrix is from the center relative to the focus.

Worked Example

Problem: Find the directrix of the parabola y=18x2y = \frac{1}{8}x^2.
Step 1: Write the equation in the form y=14px2y = \frac{1}{4p}x^2 and solve for pp.
14p=18  ⟹  4p=8  ⟹  p=2\frac{1}{4p} = \frac{1}{8} \implies 4p = 8 \implies p = 2
Step 2: The directrix of a vertical parabola with vertex at the origin is the line y=−py = -p.
y=−2y = -2
Answer: The directrix is the line y=−2y = -2. The focus is at (0,2)(0, 2), and every point on the parabola is equidistant from (0,2)(0, 2) and the line y=−2y = -2.

Why It Matters

The focus-directrix property is the unified definition behind all conic sections, which you will use throughout precalculus and analytic geometry. Satellite dish and antenna designs rely on the directrix relationship to position receivers at the focus of a parabolic reflector.

Common Mistakes

Mistake: Placing the directrix on the same side as the focus.
Correction: The directrix is always on the opposite side of the vertex from the focus. For y=14px2y = \frac{1}{4p}x^2 with p>0p > 0, the focus is above the vertex at (0,p)(0, p) and the directrix is below at y=−py = -p.