Parametrize intersection of a plane and paraboloid

[re12]

Homework Statement

Parametrize the intersection of
the paraboloid z = x² + y²
and the plane 3x -7y + z = 4
between 0 \leq t \geq 2*pi

When t = 0, x will be greatest on the curve.

Homework Equations

The Attempt at a Solution

I never really know how to do these kinds of problem. I am more familiar with parametrizing straight lines. Here is what I have done so far

I substitute the z in the plane equation with the paraboloid

3x - 7y + x² + y² = 4
x² + 3x + (3/2)² + y² -7y + (7/2)² = 37/2
(x + 3/2) ² + (y - 7/2)² = 37/2

which is a circle centered at (-3/2 , 7/2) with radius 37/2

So to parametrize x, I did

x = \sqrt{37/2} - (3/2) at t = 0 so
x = (\sqrt{37/2} - 3/2) * cos(t)

This may be wrong, but I am not sure. Please let me know if I am on the right track and how can I continue with this problem. The y and z components seem to be more complicated.

 

[gabbagabbahey]

3x - 7y + x² + y² = 4
x² + 3x + (3/2)² + y² -7y + (7/2)² = 29/2

What happened to the 4?

So to parametrize x, I did

x = \sqrt{29/2} - (3/2) at t = 0 so
x = (\sqrt{29/2} - 3/2) * cos(t)

Close, to find x, you start at the center (-3/2) and add some fraction of the radius...that means you have x=-3/2+f(t)*radius, and so you want x=-3/2+cos(t)*radius not cos(t)*(-3/2+radius)...make sense?

 

[re12]

I added (3/2)² and (7/2)² to both side so it will be greater than 4. I think I put in the wrong numbers when I use my calculator. It should be 37/2 instead of 29/2
And that explanation made a lot of sense heh.

So I ended up with
x = -3/2 + (\sqrt{37/2}*cos(t)
y = 7/2 + (\sqrt{37/2}*sint(t)

Can anyone lead me on the right track to finding z? Thanks

 

[gabbagabbahey]

z=x^2+y^2

 

[re12]

Love it whenever a problem that looks complicated has simple solution. heh thanks =)