1. The problem statement, all variables and given/known data Consider the surface z = f[itex]\left(x y\right)[/itex] = xy. Given a point P = [itex]\left[a, b, ab\right][/itex] on this surface, show that the lines with direction vectors u = [itex]\left[1, 0, b\right][/itex] and v = [itex]\left[0, 1, a\right][/itex] through P are entirely contained in the surface. 2. The attempt at a solution To be honest, I'm not really sure what I'm supposed to do here or how to go about doing it, if someone could explain it and/or give me a starting point, that'd be very helpful. 1. The problem statement, all variables and given/known data Consider the curve given by x3 + y3 = 3axy. What happens as t[itex]\rightarrow[/itex]-1? Consider the limit as t[itex]\rightarrow[/itex]-1 of x(t) + y(t). What do you conclude? The curve has obvious symmetry. Verify this using your parametrization. 2. Relevant equations The parametrized equations are: x = 3at / 1 + t3 y = 3at2 / 1 + t3 2. The attempt at a solution For the limit, after adding and simplifying the equations, it equals -a. I believe the equation is asymptotic at t = -1, but I guess I'm not quite sure what I'm supposed to conclude or what -a means. For the symmetry part, it is symmetric about y = x, so that for every point (x, y) there is a point (y, x). How do I show this with the parametric equations?