A circular paraboloid can be a elliptic paraboloid?

Question

I'm aware of this similiar question:

what is the difference between an elliptical and circular paraboloid? (3D)

But I need help in a different way. In my calculus exam, I was asked to name the following surface:

$$(x-2)^2+(y-4)^2+1=z$$

I named it as an elliptic paraboloid, and according to my teacher, it is wrong because the correct answer is circular paraboloid.

What I was thinking at that time was that it can't be wrong to call elliptic paraboloid since horizontal planes would give a circumference which is a specific case of a ellipse.

Later I questioned why my question was wrong and explained the above. She gave me an explanation that I really couldn't understand at the time, but it was short and not convincing.

So I'm searching for definitions of a elliptic paraboloid and the "best" thing I found was this from a dictionary (not good I know, that's why I need your help, please):

http://www.thefreedictionary.com/Elliptic+paraboloid

Is it going to be convincing? How can I argue in my case?

Calculus is really important to my course and I want to do good. Thanks in advance, sorry my english.

Answer 1

A circle is a special case of an ellipse; a circular paraboloid is a special case of an elliptic paraboloid. This is how I use these terms, and so do many others.

But it's possible that your teacher used them differently, when introducing these terms in the context of the class. That's her decision, and it's natural that she expects answers to be consistent with the terminology adopted for the class.

Also, even when circular is considered a special case of elliptical paraboloid, there may still be an expectation that a student will pick the answer that is the best description of the object. The more specific description "circular" conveys more information and is to be preferred.

Answer 2

$$(x-2)^2+(y-4)^2=a^2$$

is a circle with radius $a$ whose center is at $2,4$

$$(x-2)^2+(y-4)^2+1=z$$

is a circular paraboloid with whose center is at $2,4$

Instead of looking at coefficients of $x^2, y^2$ ( which must be equal for circularity ) you are looking at other displacement coefficients that are unequal.