Homework SourseLet C be the curve of intersection of the plane xy2z2 and th
So of course I have absolutely no resources on how to solve this. All my examples show intersections between a cylinder and a plane. I have no idea what the x(t) and y(t) are equal to. I mean I know they will be a circle, but the radius of the circle will increase as z increases, so I don\'t think I can just put x=cost and y=sint. I\'m just sick of spending 20+ hours on math assignments and getting absolutely nowhere. I wish textbooks would show difficult examples rather than the shortest ones.
Solution
The solution that occurs to me immediately is to solve for z in terms of x and y in the first equation and substitute into the third.
x + y + 2z = 2, so 2z = 2 - x - y, or z = 1 - x/2 - y/2
Then, we may substitute into the second equation.
This gives us 1 - x/2 - y/2 = x2 + y2 , or
x2 + x/2 + y2 + y/2 = 1
Completing squares, we have (x + 1/4)2 - 1/16+ (y + 1/4)2 - 1/16 = 1 , or
(x + 1/4)2 + (y + 1/4)2 = 18/16 = 9/8
This is a circle centered at -1/4, -1/4 with radius3/(22) = 32/4
As we know, the parametrization of a unit circle is (cos t, sin t)
Thus, the parametrization of this circle is (-1/4 + 32/4 cos t, -1/4 + 32/4 sin t)
As z = 1 - x/2 - y/2, z = 1 - (-1/4 + 32/4 cos t)/2 - (-1/4 + 32/4 sin t)/2 =
1 + 1/8 - 32/8 cos t + 1/8 - 32/8 sin t = 5/4 - 32/8 cos t - 32/8 sin t
Thus, the complete parametrization is
(-1/4 + 32/4 cos t, -1/4 + 32/4 sin t, 5/4 - 32/8 cos t - 32/8 sin t ) for 0 <= t < 2
We can plug this into the 2 equations to verify.
First equation: Show x + y + 2z = 2
x + y + 2z =
-1/4 + 32/4 cos t -1/4 + 32/4 sin t + 2(5/4 - 32/8 cos t - 32/8 sin t) =
-1/4 + 32/4 cos t -1/4 + 32/4 sin t + 5/2 - 32/4 cos t - 32/4 sin t =
-1/2 + 5/2 =
4/2 =
2
Second equation:
As x2 + y2 = z, we need to show that
x2 + y 2 - z = 0
(-1/4 + 32/4 cos t) 2 + (-1/4 + 32/4 cos t) 2 - (5/4 - 32/8 cos t - 32/8 sin t) =
1/16 + 9/8 cos2 t - 32/8 cos t + 1/16 + 9/8 sin2 t - 32/8 sin t - 5/4 + 32/8 cos t + 32/8 sin t =
1/16 + 1/16 + 9/8(cos2t + sin2t) - 5/4 = (as cos2t + sin2t = 1)
2/16 + 9/8 - 5/4 = 1/8 + 9/8 - 5/4 = 10/8 - 5/4 = 5/4 - 5/4 = 0
