A. Let L = {(x,y,z)|the distance from (x,y,z) to the y-axis is 6}. Describe what shape L is.
So far, I have that it's a circle, but I'm not sure how to describe it fully. Would it be a circle that has a radius of 6?
Let L = {(x,y)|the distance from (x,y) to the y-axis is 6}.Describe the shape of L.
I was just wondering, why are the answers to this problem and the one before it different?
B. Let L = {(x,y,z)|(x,y,z) is equidistant from (0,0,4) and the plane z=0}. Describe the shape of L using symmetry and logic.
I understand doing this by algebra, but how would the shape be explained using symmetry?
For part A:
The first shape is a circular cylinder of radius 6 whose axis (centerline) is the $y$ axis.
The second one is a circle of radius 6, lying in the $xy$-plane, with center at the origin.
The circle is the intersection of the cylinder and the $xy$-plane (i.e. the plane $z=0$).
For part B:
Think about what's happening in the $xz$ plane. The shape is the set of points that are equidistant from the point $(0,0,4)$ and the line $z=0$. You are supposed to know (I guess) that this is a parabola. By symmetry, the 3D shape is a paraboloid (which you get by rotating the parabola around the $x$ axis).