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I'm not wondering what the formula is—I already know that. For a parabola in standard form of $(x-h)^2=4p(y-k)$ I know that the focal width is $|4p|$.

But what does that mean, conceptually?

What does that distance, $|4p|$, represent? If I were to graph the parabola, would that distance be some measurable value between the focus and something else? Or between the vertex and something else? It's easy enough to solve what the focal width is; I just want to know what the point of it is. Can someone please explain focal width, as a concept, in plain English?

soapergem
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3 Answers3

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This is the length of the focal chord (the "width" of a parabola at focal level).

Let $x^2=4py$ be a parabola. Then $F(0,p)$ is the focus. Consider the line that passes through the focus and parallel to the directrix. Let $A$ and $A'$ be the intersections of the line and the parabola. Then $A(-2p,p)$, $A'(2p,p)$, and $AA'=4p$.

user35603
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(In plainer English) Imagine a regular $x^2$ parabola. It is facing up, and the vertex is at $(0,0)$. Now, imagine a line parallel to the directrix (and in this case, the $x$ axis) that runs through the focus of the parabola. This line intersects the parabola at two points; one on either side of the focus. The distance between these points is the focal width (which is $4p$). So, the focal width can be defined simply as the distance between the two arms of the parabola when they have the same $y$ value as the focus.

A charted parabola

user35603
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Cooper
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It's how wide the parabola is at the focus. Simple as that.

Bill
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  • Okay, but the mathematical meaning can be stated a bit more clearly. The width is the length of the secant line segment through the parabola's focus parallel to the directrix, also known as the latus rectum. – hardmath Dec 15 '14 at 17:33