Circle radius 0, algebraic manipulation

[1MileCrash]

Homework Statement

Going over some old tests, I am asked to find the contour of the function:

T = 100 - x^2 - y^2

at T = 100, T = 0, etc.

I have a question regarding the contour at T = 100

Homework Equations

The Attempt at a Solution

Consider T = 100

100 = 100 - x^2 - y^2
0 = -x^2 - y^2

This is a circle with radius 0. I knew at the time that this must be the top of the paraboloid. However, I also noted that

0 = -x^2 -y^2
0 = x^2 + y^2
y^2 = -x^2
y = sqrt(-x^2)

Is it correct to determine this result as "there exists a real solution only at x = 0, thus the only point at this contour is (0,0)"?

 

[HallsofIvy]

In fact, $$100- x^2- y^2= T$$ is the same as $$x^2+ y^2= 100- T$$ which, for T< 100, is the equation of a circle, with center at (0, 0) and radius $\sqrt{100- T}$. When T= 100, that reduces to $$x^2+ y^2= 0$$ which is, as you say, only true for (x, y)= (0, 0). The contour is a single point (which you can think of as a circle with radius "0").