Find a parametric representation for the curve x^2 + y + z = 2 and xy + z = 1.

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- Oct 8th 2008, 10:03 AMsuperfetsparametric representation of a curve
Find a parametric representation for the curve x^2 + y + z = 2 and xy + z = 1.

- Oct 8th 2008, 12:52 PMOpalg
This looks a bit tricky, because the curve splits into two separate pieces. If you subtract the sescond equation from the first then you get $\displaystyle x^2+y-xy=1$, which factorises as $\displaystyle (x-1)(1+x-y)=0$.

So the curve consists of the two pieces $\displaystyle x=1,\ y+z=1$ and $\displaystyle y=x+1,\ z=1-x-x^2$. It's easy enough to parametrise the two pices separately, and that is presumably the best that can be done. - Oct 8th 2008, 01:21 PMsuperfetsthanks for the effort
Don't think that's what i'm looking for. The first eq'n is a surface (parabolic cylinder) and the second eq'n is a plane. The intersection of the surface and the plane in 3 dimensions will give you a curve. It can't be in 2 pieces. I need to describe this curve by the three parametric functions. i.e. C; x = x(t) , y = y(t) , z = z(t). But thanks for trying.

- Oct 8th 2008, 03:26 PMmr fantastic
- Oct 9th 2008, 12:01 AMOpalg
The second equation is certainly not a plane. The equation xy + z = 1 is not linear. It represents a hyperbolic surface which contains lines, including the line parametrised by (x,y,z) = (1,t,1-t) that I indicated in my previous comment.

You can also see this by following**mr fantastic**'s method and paying particular attention to his final paragraph (look carefully at what happens at the particular value t=1 in his parametrisation).