I am facing the following problem.

Let’s consider 2 points that are not known (x_{0},y_{0}) and (x_{1},y_{1})

I know that from these 2 unknown points (x_{0},y_{0}) and (x_{1},y_{1}) a number of second degree polynomials passes.

f_{i}(x)=a_{i2}x^{2}+a_{i1}x+a_{i0}

For each of these polynomials I know

one point (x_{ki},y_{ki}) of the polynomial and its leading coefficient i.e. a_{i2}which is different for each polynomial.

Is it possible to find the intersection points (i.e. the 2 unknown points) of the aforementioned polynomials?