<p>We present the ways of constructing special subsets of conics present in the pencils of conics using Geometric Algebra for Conics (GAC). In particular, we offer geometrically oriented approaches to obtain the line-pairs and generalised parabolas that can be found in the pencils, by applying the tools of GAC and the classical theory of projective conics. In addition, we also describe the construction of a conic passing through five points and, throughout the work, we demonstrate the usage of points at infinity in the mentioned problems as well. The text is accompanied by examples with corresponding figures and includes a partial classification of some of the cases one may encounter in the topic.</p>