Focus and Directrix
In short: Focus and directrix form a matched pair, one anchor point plus one anchor line, and a parabola collects every spot that is equally far from the pair.
GeometryGeometryG-GPE.A.2
What is Focus and Directrix?
The focus and directrix are a lamp-like point and a wall-like line, and the parabola is the curve of every spot exactly as far from the lamp as from the wall.
The turning point of the curve sits midway between the two, a distance \(p\) from each. With that turning point at the origin and the curve opening upward, the focus is \((0, p)\), the directrix is \(y = -p\), and the equation reads \(x^2 = 4py\). Satellite dishes and headlights exploit this to gather or send out parallel rays.
How to spot it
- A problem says every point on a curve is the same distance from a point and a line.
- A parabola is described by its focus and directrix instead of by a formula.
- A satellite dish, spotlight or telescope mirror has a receiver placed at a special point.
- An equation like \(x^2 = 8y\) is given and the location of the focus or directrix is requested.
The method
- Locate the vertex halfway between the focus and the directrix and compute the gap \(p\).
- Decide the opening direction: toward the focus, away from the directrix.
- Write the equation, such as \((x – h)^2 = 4p(y – k)\) for an upward opening with vertex \((h, k)\).
- Verify a point: its distance to the focus must equal its distance to the directrix.
Focus and Directrix: solved example
Problem
A parabola has focus \((2, 5)\) and directrix \(y = 1\). Write its equation and verify that \((6, 5)\) lies on it.
Solution
The vertex is halfway between the focus and the line: \((2, 3)\). The gap is \(p = 5 – 3 = 2\), and the parabola opens upward toward the focus.
Equation: \((x – 2)^2 = 4(2)(y – 3) = 8(y – 3)\). Check \((6, 5)\): left side \(16\), right side \(8 \times 2 = 16\). Distances: to the focus 4, to the directrix \(5 – 1 = 4\). The equation is \((x – 2)^2 = 8(y – 3)\).
Try it
Question 1
Find the focus and directrix of \(x^2 = 16y\).
Show the answer
\(4p = 16\), so \(p = 4\). Focus \((0, 4)\), directrix \(y = -4\).
Question 2
A mirror curve has its focus at \((0, -3)\) and its directrix along \(y = 3\). Write its equation.
Show the answer
The vertex is the origin and \(p = -3\) because the curve opens downward. The equation is \(x^2 = 4(-3)y = -12y\).
Focus and Directrix vs. Vertex of a Parabola
The vertex is the turning point of the curve, the single spot where it is closest to the directrix. The focus is a different point, inside the curve and at distance \(p\) from the vertex, and the directrix is a line outside it. The vertex sits at the midpoint between those two.
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