# Name for constant distance in ellipse and hyperbola

Let $F_1$ and $F_2$ be two fixed points (the foci).

All the points $P$ such that $\overline{PF_1}+\overline{PF_2}$ remains constant will form an ellipse.

All the points $P$ such that $\overline{PF_1}-\overline{PF_2}$ remains constant will form an hyperbola.

Is there a name for the constants $\overline{PF_1}\pm\overline{PF_2}$?

I should perhaps add that for horizontal ellipses, that constant's value is $2a$ and for vertical ellipses, it is $2b$ if the ellipse's equation is:

$\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$

I would like to refer to that constant without having to explain everytime where it comes from.

• In an ellipse $PF_1+PF_2$ is the length of the major axis. In a hyperbola $|PF_1-PF_2|$ is the distance between the vertices. – Jack D'Aurizio Jan 5 '18 at 23:47
• True, but I am more interested in the name of the geometrical construction ($\overline{PF_1}$ added or subtracted to $\overline{PF_2}$) rather than its value. I don't want to refer to "the major axis", but really the segments described above taken together. – orion2112 Jan 5 '18 at 23:56
• You asked for the name of the constant. "The length of the major axis" is a perfectly good name for the constant. If you want a name for the union of the two line segments, then such a geometrical object is obviously not constant. – Rahul Jan 6 '18 at 0:26
• @Rahul Sorry if I was unclear. What I was looking for was something shorter than that. For circles, instead of saying "length between center and a point" we have "radius". Now does the path from $F_1$ to $P$ to $F_2$ (where $P$ is an arbitrary point of the ellipse) have any name? It shares the same length as the major axis, but it isn't the major axis, so from this point of view, the constant $2a$ or $2b$ could possibly have another name, which is the one I'm looking for if it exists. If not, then that's all I wanted to know! – orion2112 Jan 6 '18 at 3:40
• @orion2112: I'm seeing (and using) the term "major/minor radius" (instead of "major/minor semi-axis") more and more in discussions of ellipses. I haven't actually seen "transverse/conjugate radius" for hyperbolas, but they don't seem unreasonable to me. (Well, "conjugate radius" seems a little odd.) So, "major diameter" and "transverse diameter" may be appropriate to your needs. As a catch-all term for ellipses and hyperbolas, you might opt for "primary radius/diameter" or perhaps "vertex radius/diameter". – Blue Jan 6 '18 at 6:58