In Parabola Problem, Michael introduces the problem of finding a conic section given 5 of its points.
The conic section will have the form
where the constants are determined up to multiplicity. WLOG, we may set . This is why we have a well determined system of 5 equations and 6 unknowns. This then becomes an ugly (to me) system of equations to solve.
I made the observation that we can just use the equation:
Why does this work? What is the more general principle that is applied in this scenario?
Hint: What is the matrix three point form of a plane? Given 3 points in 3 dimensions in general position (aka not all 3 points lie on the same line), the unique plane that passes through is
This generalizes to dimensions.