# Thread: Sketching Surfaces (Quick and Easy Question)

1. ## Sketching Surfaces (Quick and Easy Question)

Why is that,

$x= \sqrt{y^2 + z^2}$

Is a single cone while,

$x^2 = 9y^2 + z^2$

consists of 2 cones, tip to tip?

I know that the second surface consists of elliptical cones, but I don't see how that could make any difference.

How come for,

$x= \sqrt{y^2 + z^2}$

my solutions manual only shows one cone? Couldn't I rewrite this as,

$x^{2} = y^{2} + z^{2}$

Certainly there are negative values of x that will satisfy this equation as well, right?

2. right (0,0,0) is the vertex of the cone. And {x<0} is another branch of the second cone.
While for the first one only the {x>0} branch is allowed.

3. Originally Posted by xxp9
right (0,0,0) is the vertex of the cone. And {x<0} is another branch of the second cone.
While for the first one only the {x>0} branch is allowed.
I can see that for the first one only x>0 is allowed, but why?

4. Originally Posted by jegues
I can see that for the first one only x>0 is allowed, but why?
Becaues for real numbers, all functions are single valued. $\sqrt{a}$ is defined as the non-negative number x such that $x^2= a$. $\sqrt{4}= 2$, not -2. Square roots are always non-negative.

5. Originally Posted by HallsofIvy
Becaues for real numbers, all functions are single valued. $\sqrt{a}$ is defined as the non-negative number x such that $x^2= a$. $\sqrt{4}= 2$, not -2. Square roots are always non-negative.
In which case or scenario do we end up seeing the plus or minus case then?