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If we take the zero locus of $x^2+y^2-z^2$ to be our cone, I'd like to know how to go about finding the intersection of the cone and a generic plane $Ax+By+Cz+Dw=0$.

The result will be a conic, but what are the steps needed to get there?

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In what form would you expect your result to be? In other words, how would you want to describe the conic, apart from the obvious solution “the set of all points fulfilling both equations”? Note that you should perhaps homogenize the equation of the cone. –  MvG Jul 9 '12 at 12:41

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