# Relationship between 2 Dimensional Quadratic systems and roots

Given four points

$(x_1, y_1) (x_2, y_2) (x_3, y_3) (x_4, y_4)$

How does one construct a system of two equations:

$a_1x + a_2x^2 + a_3y + a_4y^2 + a_5xy = c_1$

$b_1x + b_2x^2 + b_3y + b_4y^2 + b_5xy = c_2$

such that the set of solutions of this system is the four original points?

Solving the general systems is growing massively complicated.

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Call the four given points $P_1, P_2, P_3, P_4$. Let $\ell_{ij}$ denote the line connecting $P_i$ and $P_j$, and $L_{ij}$ a linear equation in $x$ and $y$ satisfied by the points on $\ell_{ij}$. Then you can use the system of equations: $$L_{12} L_{34} = 0, \\ L_{13} L_{24} = 0.$$
Indeed, it is easy to see that each of the four points $P_i$ satisfy both equations, so they lie on the intersection. Provided none of the lines are identical (the condition that the points are in general position), then there are exactly four solutions to this system of equations, which you've already found.