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We study quotients of quadratic forms and associated polar lines in the projective plane. Our results, applied pointwise to quadratic differential forms, shed some light on classical binary differential equations (BDEs) associated to congruences of lines in Euclidean 3-space and allows us to introduce a new one. The new BDE yields a new singular surface in the Euclidean 3-space associated to a congruence of lines. We determine the generic local configurations of the above BDEs on congruences.

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We study in this paper local codimension 2 singularities of (first order) implicit differential equations F (x, y, p) = 0, where F is a germ of a smooth function, p = dy dx , F p = 0 and F pp = 0 at the singular point. We obtain topological models of these singularities and deal with their bifurcations in generic 2-parameter families of equations.