- #1

dr721

- 23

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## Homework Statement

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.

## Homework Statement

Consider the curve given by x

^{3}+ y

^{3}= 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.

## Homework Equations

The parametrized equations are:

x = 3at / 1 + t

^{3}

y = 3at

^{2}/ 1 + t

^{3}

**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?