Let $$ax^2+by^2+cz^2+2dyz+2exz+2fxy=0 \\ gx^2+hy^2+jz^2+2kyz+2lxz+2mxy=0$$ be two conics in barycentric coordinates. Can you suggest a way or algorithm to find the equation of a line passing through the point of intersection of two conics?
I tried solving but this way not giving any progess
Can anybody suggests the way to deal this situation.
For convenience I will use ordinary coordinates, WLOG.
The linear combination
$$(1-\lambda)(ax^2+2fxy+by^2+2dy+2ex+c)+\\ \lambda(gx^2+2mxy+hy^2+2ky+2lx+j)=0$$
describes a pencil of conics which have the same four points in common. For three particular values of $\lambda$ the conic degenerates to a pair of straight lines ($(AB,CD),(AC, BD),(AD,BC)$). It takes a cubic equation to obtain these values.
Alternatively, by a suitable combination of the two equations, you can eliminate the terms in $y^2$ and express $y$ in terms of $x$. Then plugging $y$ in one of the initial equation, you will end-up with a quartic equation, which is solvable analytically.
Note that in the classical resolution method of the quartic, you factor as the product of two quadratics, by means of a resolvent equation of the third degree. These just correspond to the degeneracy equation and pairs of straight lines as in the pencil method.
