Degenerate conic  In geometry, a degenerate conic is a conic (a second-degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible curve. This means that the defining equation is factorable over the complex numbers (or more generally over an algebraically closed field) as the product of two linear polynomials. Using the alternative definition of the conic as the intersection in three-dimensional space of a plane and a double cone, a conic is degenerate if the plane goes through the vertex of the cones. In the real plane, a degenerate conic can be two lines that may or may not be parallel, a single line (either two coinciding lines or the union of a line and the line at infinity), a single point (in fact, two complex conjugate lines), or the null set (twice the line at infinity or two parallel complex conjugate lines). All these degenerate conics may occur in pencils of conics. That is, if two real non-degenerated conics are defined by quadratic polynomial equations and , the conics of equations form a pencil, which contains one or three degenerate conics. For any degenerate conic in the real plane, one may choose and so that the given degenerate conic belongs to the pencil they determine. The conic section with equation formula_1 is degenerate as its equation can be written as formula_2, and corresponds to two intersecting lines forming an "X". This degenerate conic occurs as the limit case formula_3 in the pencil of hyperbolas of equations formula_4 The limiting case formula_5 is an example of a degenerate conic consisting of twice the line at infinity. Similarly, the conic section with equation formula_6, which has only one real point, is degenerate, as formula_7 is factorable as formula_8 over the complex numbers. The conic consists thus of two complex conjugate lines that intersect in the unique real point, formula_9, of the conic. The pencil of ellipses of equations formula_10 degenerates, for formula_5, into two parallel lines and, for formula_3, into a double line. The pencil of circles of equations formula_13 degenerates for formula_14 into two lines, the line at infinity and the line of equation formula_15. Over the complex projective plane there are only two types of degenerate conics – two different lines, which necessarily intersect in one point, or one double line.