Focus and Directrix

Algebra 2 · Axiom Academy

The geometric definition of a parabola: the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). Let's start by identifying these two components. In the animation the focus sits above the directrix, and a test point P is measured against both: its distance to the focus, and its perpendicular distance down to the directrix. Most points are closer to one than to the other — the parabola is made of exactly the points where the two distances tie . For any point on the parabola, if we measure the distance to the focus and the perpendicular distance to the directrix, these distances are equal . Here the parabola is drawn with focus F(0, 1) and directrix y = -1 — watch the two readouts tick in lockstep as P slides along the curve. This is the defining property! At every position, the slanted distance from P to the focus F agrees exactly with the straight drop from P to the directrix. On , take the point (2, 1) . Its distance to the focus F(0, 1) is , and its perpendicular distance to the line y = -1 is 1 - (-1) = 2 . Equal — exactly as the definition demands. Let's trace out the parabola using nothing but the equidistant property. Starting at the vertex and growing outward in both directions, every stamped point keeps its two distances equal — watch each frontier point carry its matching pair of measurements as the curve assembles.

This is the written version of the interactive lesson above. See the full Algebra 2 course.