On conics and their tangents

We present, in a way quite accessible to undergraduate and graduate students, some basic and important facts about conics: parabola, ellipse and hyperbola. For each conic, we start by its definition, then consider tangent line and obtain an elementary proof of the reflexion property. We study intersection of tangents. We obtain the orthopic set for orthogonal tangents: the directrix for parabola and the Monge’s circle for ellipse and hyperbola. For ellipse and hyperbola we also consider intersection of tangents for parallel rays at points of intersection with the conic. Those analysis lead to geometric methods to draw conics. Finally we get the directrices for ellipse and hyperbola by considering intersections of tangents at endpoints of a secant passing through a focus.