Equation of the Conic through Five Given Points

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There are a family of conics that go through four points; the fifth point defines one conic. One way to determine this conic is to first make a family for four points. A numeric example is shown below.

If we draw the quadrilateral joining the four points and if its sides in order have equations α=0, β=0, γ=0, δ=0, then the equation
αγ+kβδ=0 is the equation of a conic for any value of k, and goes through all four points.