True or False: A line is a parabola.

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

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