Suppose we have parabola $y=ax^2+bx+c$, which has focus at $(-\frac b{2a},\frac 1{4a}-\frac {b^2}{4a}+c)$. There is a line $\ell$ at $y=\frac{a^2-b^2}{4a}+c$ which has the following property: any two lines $m,n$ which pass through consecutive pairs of equally-spaced points will intersect $\ell$ a constant distance from each other, dependent only upon the spacing of the points chosen for determining $m,n$.

More specifically, if $m$ passes through points of our parabola at $x_0,x_1$ and $n$ passes through points of the parabola at $x_1, x_2$ and $x_1-x_0=x_2-x_1=q$ for some constant $q$, then there exists a constant $r$ such that regardless of how $x_0$ is chosen (with $x_1 = x_0+q,x_2 = x_0+2q$), we have that the intersection at $\ell,m$ is distance $r$ from the intersection at $\ell, n$.

The math gets a bit messy, but the surprise to me was that this number ${a^2-b^2\over 4a}+c$ was only different from the "height" of the focus of the parabola by $\frac a4-\frac 1{4a}$.

Is this quantity ${a^2-b^2\over 4a}+c$ a well-known part of conic theory? Does it have a name? Does it have any more general usage or properties with other conic sections?

Side-note: at first I almost believed that this quantity was the same as the value of the "height" of the focus since I started by looking at parabolas with $a=1$...

  • $\begingroup$ Unfortunately, it is not the directrix. That would've been really cool. You may have stumbled upon something new. $\endgroup$ – GaussTheBauss Feb 25 '16 at 23:48

I think I know what you are talking about. But it is not a uniquely defined line. With the right parameters it could actually be any line that crosses the given parabola and not parallel to the axis.

Think of the tangents to the parabola, which I shall call P. We select any one tangent T. Assuming that the x-axis is taken to be parallel to the directrix of P, any other tangents drawn from points on P with equally spaced x-coordinates will intersect T at equidistant points. The x-coordinates of the intersection points will have half the spacing of the x-coordinates of the tangency points. This applies to any tangent we might choose for T; try it!

Now suppose we draw secants to P in the manner described by the OP. These will be tangent to a second parabola Q, whose own points will lie on the same side of P as the focus does. The equidistant-intersection line is then any tangent of Q. Moreover, the location of Q itself will depend on the spacing we choose for the x-coordinates on P.

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  • $\begingroup$ Thank you; perhaps I'll clarify that I intended the line parallel to the directrix, or maybe I'll leave it open as it is and see what else comes up. I guess this also means that by choosing $q=1$ I missed a lot of different possible lines $\ell$. $\endgroup$ – abiessu Feb 26 '16 at 5:13
  • $\begingroup$ You're welcome. As you now see, even with lines parallel to the directrix the answer of still not unique. You may want to explore what happens when you "fit" the x-coordinate spacing q to the intrinsic characteristics of the parabola by setting q to four times the distance from the vertex to the focus. This q is the length of a chord perpendicular to the axis and passing through the focus, it's called the latus rectum. $\endgroup$ – Oscar Lanzi Feb 26 '16 at 9:37

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