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points) are necessary to specify a conic, as each constraint cuts the dimension of possibilities by 1, and one starts with 5 dimensions;
* in 5 dimensions, the intersection of 5 (independent) hyperplanes is a single point (formally, by Bézout's theorem);
* general linear position of the points means that the constraints are independent, and thus do specify a unique conic;
* the resulting conic is non-degenerate because it is a curve (since it has more than 1 point), and does not

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