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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.

Thanks again!