# parametrize intersection of a plane and paraboloid

by re12
Tags: intersection, paraboloid, parametrize, plane
 P: 3 1. The problem statement, all variables and given/known data 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. 2. Relevant equations 3. 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.
HW Helper
P: 5,004
 Quote by re12 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?
 P: 3 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
HW Helper
P: 5,004

## parametrize intersection of a plane and paraboloid

z=x^2+y^2
 P: 3 Love it whenever a problem that looks complicated has simple solution. heh thanks =)

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