Parametric equations for curves of intersection and their direction
See figure attached for problem statement, as well as my attempt.
I'm just curious as to whether or not I've done this problem correctly, and whether or not I will ever run into trouble doing problems this way.
I say this way, because I don't even concern myself with what the actual curve looks like.
They always give some sort of information to establish the desired direction they want the curve to go in, and I just work from that.
In this example they say, from a point far up the positive z-axis.
So I draw my point of view from a point far up the positive z-axis and choose a simple point that will be on curve. In this case I chose when y=0.(i.e. the point on the x-axis)
Once I have this point I simply think about which way the curve will progress if I increase my value for t. If it's going in the wrong direction, I simply replace all the t's in my parametric equations by (-t), thus reversing the direction.
Am I safe solving problems this way? As I said previously, I never actually draw or make a rough sketch of what the curve of intersection looks like, but do I really need to?
Hopefully my work is correct, and if someone could clarify on the validity of my procedure that'd be great.