Parameterization, Folium of Descartes, etc.

[dr721]

Homework Statement

Consider the surface z = f\left(x y\right) = xy. Given a point P = \left[a, b, ab\right] on this surface, show that the lines with direction vectors u = \left[1, 0, b\right] and v = \left[0, 1, a\right] 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.

Homework Statement

Consider the curve given by x³ + y³ = 3axy.

What happens as t\rightarrow-1? Consider the limit as t\rightarrow-1 of x(t) + y(t). What do you conclude?

The curve has obvious symmetry. Verify this using your parametrization.

Homework 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?

 

[dr721]

Sorry for the poor formatting. I don't know how to format column vectors or fractions.