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This was originally a homework problem but I realized my misunderstanding stems from the explanation given before the problem so here we are. The thread deals with these 3 pages from Spivak's Calculus:

https://ibb.co/kAKyVU

https://ibb.co/jXVSPp

https://ibb.co/kwRdVU

I'm pretty sure this is copyright infringement but quoting all this without being able to show the illustrations would just serve to confuse anyone who reads it, so i'll take down the links after I figured things out.

So here's my problem: In page 81 the author claims that after combining equations (1) and (2) on can conclude that a point ##(x,y,z)## is in the intersection between the plane and the cone if and only if ##Mx+B=±C\sqrt{x^2+y^2}##. My first problem comes from this equation; although every point in the intersection adheres to this equation, the opposite is not true. You could find infinite points with ##x## and ##y## coordinates that adhere to this equation but with some random ##z## coordinate that are not in the intersection. The proper equation (or system of 2 equations) to my understanding should be ##Mx+B=±C\sqrt{x^2+y^2}=z##.

After developing this equation, Spivak changes the coordinate system to one in the intersecting plane and somehow we end up with the final equation of the 2d intersection. So it seems to me that from equations (1) and (2) to equation (*) Spivak somehow went from the 3d space to the 2d plane and didn't bother telling about it. This just makes me more confused because the equation (*) is still expressed in terms of the original ##x## and ##y## coordinates, but the intersection is not in the ##[x,y]## plane, so I don't get how you would describe it using a 2d coordinate system when it is not in that system.

I'm sorry if this all sounds messy and unclear, since I don't really know where my misunderstanding comes from myself it's hard to express it in writing. I just hope I articulated my problem quickly enough to receive some clarification. Thanks for reading all this.