Math 2A Vector Calculus Homework Exercise 10.1#33
Find a vector function that traces the intersection of the surface of the cone with the plane .
A common approach to such a problem is to choose one of the rectangular coordinates to be the parameter and then solve the other two in terms of that one. Since there’s a nice linear relation between y and z, that looks like a good place to start.
Set
y = t. Then z = 1 + t and solving
the equation of the cone for x, . Thus
Notice that to get both sides of the curve and you need to plot the negative square root and the positive square root separately.
To get a better idea about how to parameterize here, a little recollection of conic sections is helpful. Observe that the plane is parallel to the side of the cone, and that such a cross-section is a parabola. Now x = 0 is the yz plane in which the vertex of the parabola must be. This is where z = y so the lowest point on parabola is at (0. , ).
To maybe better harness this symmetry, try having the vertex correspond to t = 0: means that and so .
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