Question: How can I find the intersection between a plane f(x,y) and a quadric g(x,z)?

Hello everyone, 

please need some brain power here to help me with this problem. I'm stacked here and I cannot find a solution. 

I would like to find the intersection between a plane and a quadric surfaces. Please notice that the two equations are written like f(x,y) and g(x,z) respectively. My first problem is that I cannot visualize the two surfaces correctly as implicitplot3d of the quadric surfaces displays nothing in return. 

I want then to find the centre of the ellipse and plot the point and the two surfaces all together in the same 3d plot. Somehow I canont make it. What is wrong with me? 

I am pasting here my MWE. 

Any kind of help would be really appreciated. Thank you so much. 





with( Optimization ):

my_plane := 2024.30587449691-.341275505799078*x-3.89936179341114*y;
my_quadric := -10595.4104931095+6.73749956241827*x+42.1022818380012*z-0.654818649508000e-1*x^2-0.421174257681115e-1*z^2;





my_quadric_plot_3d := implicitplot3d(my_quadric, x=-10000..10000,
                                       y = -100..100,
my_plane_plot_3d := implicitplot3d(my_plane, x=-10000..10000,
                                       y = -100..100,

my_ellipse := CompleteSquare( subs(y = solve(z = my_plane, y), my_quadric));



ellipse(e1, my_ellipse, [x,z]):
XY_centre_e1     := coordinates(center(o,e1)):
my_center_point := [XY_centre_e1[1],XY_centre_e1[2],
                       subs(x=XY_centre_e1[1], y=XY_centre_e1[2], my_plane)]:
my_center_point_plot := pointplot3d(my_center_point, color = "red"):

display(my_center_point_plot, my_plane_plot_3d, my_quadric_plot_3d);






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