Given two parallel lines, each defined using Hesse normal form find the intersection point.

Question

Given two parallel lines $ax + by = c_1$ and $ax + by = c_2$, I want to find the intersecting point of these lines. How can I calculate the points?

Answer 1

Over an arbitrary field in the projective plane the lines $ax+by-c_1z=0$ and $ax+by-c_2z=0$ intersect in the point whose coordinates are given by the cross-product of the coefficients of the lines, which gives the point $$(-bc_2+bc_1:ac_2-ac_1:0).$$