Write a computer program or solve algebraically for all the points of
intersection  (real and complex) of the two curves defined by

0 = A1 + A2*x + A3*y + A4*x^2 + A5*x*y + A6*y^2
0 = B1 + B2*x + B3*y + B4*x^2 + B5*x*y + B6*y^2

(References to correct solutions in the Literature would be
appreciated, but are  unnecessary.)

where Ai and Bi are given constants.

I think i know how to solve for all the questions. First write the two
euqations as a quadratic in y which would give

A6*y^2+(A3+A5*x)*y+(A1+A2*x+A4*x^2)=0
B6*y^2+(B3+B5*x)*y+(B1+B2*x+B4*x^2)=0

If you then multiply each of these equations once with y and once with
y^2 you can write it matrix, vey diffucult to write without equation
editor. Then you convert this problem to a quadratic eigen value
problem in x.

To solve this quadratic eigenvalue proglem, perform first order
realization and solve an eigenvalue problem using MATLAB. This would
give you all the solutions of x, both real and complex. Once you have
x you can solve for y.

If you need details I can send you the word file that has all the
details and the MATLAB code too.