# Is there a name for this quantity which is similar to the focus of a parabola?

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 $a=1$...

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

• 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 $\ell$. Feb 26 '16 at 5:13