Do these 3 systems of equations all all define the same curve?

[nickadams]

Homework Statement

Consider three systems of equations:

x^2 + y^2 + z^2 = 1
y^2 + z^2 = 1

x^2 + y^2 + z^2 = 1
x = 0

y^2 + z^2 = 1
x = 0

Which of these define the same curve and which define different ones?

Homework Equations

x^2 + y^2 + z^2 = R^2 is a sphere
x,y, or z = # is a plane
(x,y,z)^2 + (x,y,z)^2 = # is a cylinder

The Attempt at a Solution

I think they all define the same curve; here is why...

Equation 1 of the first system is a sphere centered at the origin with a radius of 1.
Equation 2 of the first system is a cylinder centered around the x axis.
The curve represented by those two equations is the points where both equations are satisfied; AKA: where they intersect. They intersect at x=0 in a circle given by y^2+z^2 = 1 .

Equation 1 of the second system is a sphere centered at the origin with a radius of 1.
Equation 2 of the second system is the yz plane at x=0.
The curve represented by those two equations is the points where both equations are satisfied; AKA: where they intersect. They intersect at x=0 in a circle given by y^2+z^2 = 1 .


Equation 1 of the third system is a cylinder centered at the x-axis with a radius of 1.
Equation 2 of the third system is the yz plane at x=0.
The curve represented by those two equations is the points where both equations are satisfied; AKA: where they intersect. They intersect at x=0 in a circle given by y^2+z^2 = 1 .




Am I right?

 

[Dick]

Yes, you are right.