2

It is known that a line is a degenerate parabola. But if asked as above, what is the better answer?

Context: This question appeared on a quiz recently given in our Precalculus class. It is not clear to me and my classmates if the answer is True or False. Our book says the following. enter image description here

Henry Razon
  • 51
  • Almost always. . – William Elliot Jul 18 '19 at 02:21
  • Why not always? – Henry Razon Jul 18 '19 at 02:23
  • Where did you get that definition? A parabola is the set of points equidistant from a point (focus) and a line (directrix). What are the focus and directrix of a line? – John Douma Jul 18 '19 at 02:26
  • All parabola are similar to one another. And lines are not similar to parabola. A degenerate parabola is not a parabola. – user317176 Jul 18 '19 at 02:39
  • @JohnDouma You could take any point on the line as "focus" and the perpendicular through that point as "directrix". But unlike a proper parabola, the "degenerate parabola" has more than one choice of "focus" and "directrix". It also fails to have other properties that all proper parabolas have. – David K Jul 18 '19 at 02:40
  • @JohnDouma, that is exactly what I was thinking. My answer is false because the focus-directrix definition of a parabola will not work for a line. The first sentence in the question above is not really a definition, but is simply a special case. – Henry Razon Jul 18 '19 at 03:34
  • @DavidK, can you cite a property of all nondegenerate parabolas that lines do not satisfy? – Henry Razon Jul 18 '19 at 03:36
  • I tend to favor "inclusive" definitions, so I'd say: True, degenerate parabolas are parabolas. That said, it's perfectly reasonable to have an "exclusive" view; indeed, I suspect that most discussions implicitly assume it. (When discussing parabolas/circles/triangles/whatever here, I almost-never even think to qualify my remarks as applying only to "non-degenerate" instances. It can be instructive to see what happens when figures degenerate, though.) In any case, anyone who ends such a conversation at "true or false" is doing an intellectual disservice to the topic and the conversers. – Blue Jul 18 '19 at 08:02
  • Related: Is the conjugate axis of a hyperbola itself a trivial hyperbola. I have some additional comments there about degeneracy. – Blue Jul 18 '19 at 08:05
  • Don't let language define reality. Let reality define language. – fleablood Jul 18 '19 at 09:01
  • 2
    @HenryRazon: The textbook explicitly states that degenerate cases are included in the classification of second-degree $xy$ polynomials. If the textbook defines a conic as the graph of such a polynomial, then the degenerate cases count. Now, if the textbook defined conics by other means, then degeneracies may-or-may-not be included for whatever reason. Even then, your instructor may have opted to promote an alternative view, and the quiz may reflect that view. (When I teach, I very often think outside-the-book.) This is really a discussion/debate you should be having with your instructor. – Blue Jul 18 '19 at 09:33

3 Answers3

2

No, in almost all contexts. Very occasionally you might encounter a family of parabolas one of which is degenerate; then it might be acceptable.

If you have been asked this question, tell us the context.

Ethan Bolker
  • 95,224
  • 7
  • 108
  • 199
1

The parabola is $y^2=4ax$, if you put $a=0$, it becomes a line $y=0$. So one may say that a line is a parabola whose length of latus-rectum is zero. This is how a parabola degenerates to a line. A line is the thinnest parabola.

Z Ahmed
  • 43,235
0

Consider the parabola $~y=a~x^2~$.

All parabolas can be rotated and translated to arrive at this form. Hence this equation covers all possible parabolas in the $2$D plane for the purpose of this problem.

Now, for a straight line to exist, we should be able to find a point where $~\frac{dy}{dx}~$ does not change.

But, $~\frac{dy}{dx}=2ax~$

The derivative is different for all points on the parabola since it is dependent on $~x~$ and there is only one $~y~$ for each $~x~$.

So we can conclude that the statement is false.

**************-

see https://www.quora.com/How-do-you-prove-that-there-are-no-straight-lines-in-a-parabola

nmasanta
  • 9,222