You have the two surfaces defined by , a cylinder, and , the "upper half" of the cone given by .

If you replace the in the first equation by , you get or , an ellipse with major semi-axis along the y axis with length 2 and minor semi-axis along the x-axis, of length . Thatisthe equation of the projection into the xy-plane.

I don't think the problem actually requires you to find parametric equations for the curve, but if you want to, a standard parameterization of the ellipse would be , . Then .

The boundaries of the first octant are the coordinate planes, x= 0, y= 0, and z= 0. If z= 0, then the first equation becomes and since we are in the first octant, x= 1. But then the second equation becomes which is impossible. The curve of intersection does not cross the xz-plane.

If x= 0, then the first equation becomes so and, since we are in the first octant, z= 2. then the second equation becomes so . The endpoint in the yz-plane is (0, 2, 2).

If y= 0, then the second equation becomes or . Then the first equation becomes so that , and the endpoint in the xz-plane is .