Solving Parametric Equation: Intersection of x^2 & x^2 + y^2 = 1

[tnutty]

Homework Statement

Show that the curve with parametric equation :

x = sint t;
y = cos t;
z = sni^2 t

is the curve of intersection of the surface z = x^2 and x^2 + y^2 = 1.

The Attempt at a Solution

From polor equation I know that x = rcos(t) and y = rsin(t);

from this we can replace x^2 + y^2 = 1. with cos(t) + sin(t) = 1

and since z = x^2, and x = rcos(t), it follows that z = r^2cos^2(t) = cos^2(t)

so we have the vector equation :

v = < cos(t) , sin(t), cos^2(t).

But this doesn't follow the question. Whats wrong?

 

[Dick]

The 't' in the polar equations is the theta coordinate in polar coordinates. Your parameter t in the parametric equations is NOT theta. Just show the two x,y,z equations are satisfied for any t.

 

[tnutty]

so could I just use, tan(theta) = y/x = sin t / cos t;
theta = tan ^-1(y/x)
= tan ^-1 sin t / cos t
= 1 / tan = cot?

 

[Dick]

You don't need theta at all. Forget theta. Just work with the original t. This is a question about x,y,z coordinates, not r,theta,z. There is no point in trying to change coordinates. Reread my post 2. Starting with the word "Just".