I was given this task and I solved it, but only after solving for $x^4+x^3$... $$x^2+y^2+4z^2=6y-4\quad \& \quad 2xy-4xz+4yz=y^2+5$$
How I tried: 1. I summed (and multiplying the second one with $-1$) the two equations, getting: $$(x-y+z)^2=-(y-3)^2 \implies y-3=0 \quad \& \quad x-y+z=0$$
I than put $y$ into the equations but in the end I got something long with $x^4$...
I know this task is not ment to solve polynomials, so I am asking for a simpler solution.
You need to be more careful. The two surfaces are $$ x^2 + (y-3)^2 + 4 z^2 = 5 \; , $$ $$ x^2 - (x-y+2z)^2 + 4 z^2 = 5 \; . $$ The first is an ellipsoid centered at $(0,3,0),$ the second is a hyperboloid of one sheet centered at the origin, but not a surface of revolution.
You get $$ (y-3)^2 + (x-y+2z)^2 = 0. $$ Then for real points, $y=3$ and $x = 3 - 2z.$ the ellipsoid gives $8 z^2 - 12 z + 4 = 0$ or $4 (2z-1)(z-1) = 0.$ The two points where the surfaces touch are $$ (1,3,1) $$ $$ \left(2,3,\frac{1}{2} \right) $$ The surfaces must be tangent at these points, so the gradients of the two defining functions should be parallel.
From the original way you wrote them the gradients are $$ (2x,2y-6,8z) \; , $$ $$ (2y-4z,2x-2y+4z, -4x+4y) $$
At $(1,3,1)$ the gradients are $$ (2,0,8), \; \; \; (2, 0, 8) \; \; . $$
At $ \left(2,3,\frac{1}{2} \right)$ the gradients are $$ (4,0,4), \; \; \; (4, 0, 4) \; \; . $$
