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
In Maple11, you can enter the following command to get the graph below:

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 .