Regarding your definition:
"Given a segment, the ellipse is the locus of points for which the product of the two slopes referred to the endpoints of the segment is a constant."I think it would read better if the constant were given first, along with the segment. Also, I believe the constant must be positive; the locus will be a hyperbola if the constant is negative. How about this?
1) Given a segment AB and constant c > 0 (c < 0), the set of points X for which the product of the slopes of line AX and BX equals c is an ellipse (hyperbola).
I'm not that thrilled with this definition, either. I think a definition should start (in this case) with "An ellipse is ...", rather than end with "...is an ellipse" unless one introduces parameters in the beginning that become part of the term being defined.
For example, one could define a (specific) circle as follows.
"Given a point P in a plane and radius r, the locus of points at distance r from P is said to be the circle with center P and radius r."
My definition in (1) doesn't do this, as no attributes of the ellipse are referred to.
How about this?
2) An ellipse (hyperbola) is the set of points X for which the product of the slopes of the two lines through X and two fixed points is a positive (negative) constant.
I'm still not sure I have it right, as mathematical grammar is so subtle. Of course, here, one must insert mental parenthesis and commas appropriately to read it as
An ellipse is {X: (slope of XA)(slope of XB) = c} for some points A and B and constant c>0. Then it is clear that A, B, and c are fixed ahead of time and the ellipse will depend on them. Qualifiers are one of the most difficult aspects of definitions for many students, I believe.
When one encounters the first X in (2) it'sounds like X is a set, not a point in a set, and it isn't clear until one reaches the second X that X must be a point. In my view this makes (2) not a very good definition either. Perhaps you have similar issues in the Spanish language.
It would be nice not to mention X at all. You didn't refer to X in your definition, but I didn't think the phrase "the product of the two slopes referred to the endpoints of the segment is a constant" was clear.
Well, what I thought would be a 2-minute post giving a clean definition turned into something else entirely. Sorry about that. Someone else want to give the definition a try?