The bundle of conics that pass through four points A, B, C and D, can be generated by means of the (degenerate) conics that are formed by joining the given points two by two by lines:

λ AB CD + AC BD=0.

   By varying the value of the parameter λ, the different conics that pass through the four points are obtained. These conics can be hyperbolic, parabolic or elliptical.

   By moving point D, you can build pencils of different conics. Try to answer the following questions. How many parabolic type conics can there be in each bundle? Can you find a pencil with all conics of elliptical type? And with all the conics of the hyperbolic type? Why?.

    If we also require that the conic pass through a fifth point P, then there will be a unique value of the parametrer λ for which this property verifies.

    You can check this fact by moving the point P and modifying the parametrer λ until you find a conic passing through P.