A directrix is a fixed straight line used to define a curve. Most often, you’ll encounter it when studying parabolas: every point on a parabola is exactly the same distance from a fixed point (called the focus) as it is from the directrix line. This equal-distance rule is what gives a parabola its shape, and the directrix plays a role in defining ellipses and hyperbolas too.
How the Directrix Defines a Parabola
You probably know parabolas as the U-shaped graphs of quadratic equations. But there’s a purely geometric way to build one. Place a point (the focus) and draw a straight line (the directrix) that doesn’t pass through it. Now find every point in the plane whose distance to the focus equals its distance to the directrix. The collection of all those points traces out a parabola.
The vertex of the parabola sits exactly halfway between the focus and the directrix. This halfway distance is commonly labeled “p.” If p = 3, for example, the focus is 3 units from the vertex on one side, and the directrix is a line 3 units from the vertex on the other side. The parabola always opens away from the directrix and toward the focus.
Finding the Directrix From an Equation
For a vertical parabola centered at the origin, the standard form is y = x²/(4p). The focus sits at the point (0, p), and the directrix is the horizontal line y = −p. A larger value of p means the parabola is wider and the directrix is farther from the vertex.
Horizontal parabolas work the same way, just rotated. The standard form becomes x = y²/(4p), with the focus at (p, 0) and the directrix at x = −p. If the parabola’s vertex is shifted to some point (h, k) instead of the origin, you apply the same logic but offset by h and k. For a vertical parabola with vertex (h, k), the directrix is the line y = k − p.
The Directrix in Ellipses and Hyperbolas
Parabolas aren’t the only curves that use a directrix. Ellipses and hyperbolas have them too, though they come up less often in introductory courses. The unifying idea across all three curves is a ratio called eccentricity (e): for any point on the curve, its distance to the focus divided by its distance to the directrix equals e.
For a parabola, the eccentricity is exactly 1, which is why the two distances are always equal. For an ellipse, the eccentricity is between 0 and 1, so each point is closer to the focus than to the directrix. For a hyperbola, the eccentricity is greater than 1, so each point is farther from the focus than from the directrix. Because ellipses and hyperbolas each have two foci, they also each have two directrices, one paired with each focus.
For an ellipse with semi-major axis a, each directrix sits at a distance a/e from the center, measured along the major axis. For a hyperbola, the same formula applies but the directrices fall between the two branches of the curve. The ancient Greek mathematician Pappus was among the first to study the focus-directrix relationship across all conic sections.
The Latus Rectum Connection
Another term that often appears alongside the directrix is the latus rectum. This is a chord that passes through the focus perpendicular to the axis of symmetry. Its length is tied directly to the geometry that the directrix helps define.
For an ellipse with semi-major axis a and eccentricity e, the semi-latus rectum (half the latus rectum) equals a(1 − e²). For a hyperbola, it equals a(e² − 1). For a parabola, the semi-latus rectum simply equals 2p, twice the distance from the vertex to the focus. These relationships matter because the latus rectum tells you how “wide” the curve is at the focus, and that width is a direct consequence of where the directrix sits.
Why the Directrix Matters in the Real World
The focus-directrix property isn’t just a textbook curiosity. It’s the principle behind parabolic reflectors used in satellite dishes, car headlights, and solar energy systems. Because of how a parabola is defined, any ray arriving parallel to the axis of symmetry bounces off the curved surface and passes through the focus. This works in reverse too: a light source placed at the focus sends out a perfectly parallel beam.
Concentrating solar power systems use this directly. Parabolic trough collectors and dish systems are shaped so that sunlight, which arrives in nearly parallel rays, reflects and converges at the focal point. The directrix doesn’t appear physically in the design, but the geometric relationship it creates with the focus is what makes the concentration of energy possible. Without the equal-distance property, the reflected rays wouldn’t all meet at a single point.