linear constraints on the (projective) linear space of conics, and hence specifies a unique conic, though this brief statement ignores subtleties.

More precisely, this is seen as follows:
* conics correspond to points in the five-dimensional projective space
* requiring a conic to pass through a point imposes a linear condition on the coordinates: for a fixed the equation is a linear equation in
* by dimension counting, five constraints (that the curve passes through five