(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Parametrize the intersection of

the paraboloid z = x^{2}+ y^{2}

and the plane 3x -7y + z = 4

between 0 [tex]\leq[/tex] t [tex]\geq[/tex] 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 + x^{2}+ y^{2}= 4

x^{2}+ 3x + (3/2)^{2}+ y^{2}-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 = [tex]\sqrt{37/2}[/tex] - (3/2) at t = 0 so

x = ([tex]\sqrt{37/2}[/tex] - 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.

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# Parametrize intersection of a plane and paraboloid

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