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Directrix — Definition, Formula & Examples

A directrix is a fixed line used to define a conic section. Every point on the curve has a specific distance ratio between the directrix and a point called the focus.

A directrix of a conic section is a fixed line such that, for any point PP on the conic, the ratio of the distance from PP to the focus to the distance from PP to the directrix equals the eccentricity ee of the conic. For a parabola (e=1e = 1), these two distances are equal.

Key Formula

distance from P to focusdistance from P to directrix=e\frac{\text{distance from } P \text{ to focus}}{\text{distance from } P \text{ to directrix}} = e
Where:
  • PP = Any point on the conic section
  • ee = Eccentricity of the conic (e = 1 for a parabola, 0 < e < 1 for an ellipse, e > 1 for a hyperbola)

How It Works

For a parabola with vertex at the origin and equation y=14px2y = \frac{1}{4p}x^2, the directrix is the horizontal line y=−py = -p, and the focus is at (0,p)(0, p). Pick any point on the parabola: its distance to the focus always equals its distance to the directrix. Ellipses and hyperbolas also have directrices, but the distance ratio equals their eccentricity ee rather than 1. In practice, you find the directrix by identifying pp from the equation of the conic.

Worked Example

Problem: Find the directrix of the parabola x2=12yx^2 = 12y.
Identify the form: Rewrite in standard form x2=4pyx^2 = 4py and match coefficients.
x2=4py  ⟹  4p=12  ⟹  p=3x^2 = 4py \implies 4p = 12 \implies p = 3
Write the directrix: For an upward-opening parabola with vertex at the origin, the directrix is y=−py = -p.
y=−3y = -3
Verify: The focus is at (0,3)(0, 3). Check the point (6,3)(6, 3) on the parabola: distance to focus =36+0=6= \sqrt{36 + 0} = 6, distance to directrix y=−3y = -3 is 3−(−3)=63 - (-3) = 6. They are equal, confirming the directrix.
Answer: The directrix is the line y=−3y = -3.

Why It Matters

Satellite dishes and parabolic mirrors are shaped so that signals hitting the surface reflect toward the focus. Engineers use the directrix to derive the exact curve shape, making it essential in antenna design and optics.

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. If the focus is at (0,p)(0, p), the directrix is at y=−py = -p, not y=py = p.