Extremal chord problem in ellipse

A new method to construct a tangent to the conchoid of Nicomedes or limaçon of Pascal curves is discussed. Some interesting properties of the cardioid curve (which is a special case of limaçon of Pascal) are investigated. The following problem is studied: "Given a line k and two points A and B on one side of k, find point C such that the sum of lengths of segments CD and CE is minimal, where D and E are intersections of line k with lines CA and CB, respectively". This problem is dual to the classic problem to find shortest segment inscribed to a given angle and passing through a given point. Part of this problem was solved and the remaining part is left as an open question. The problem to find ellipse's longest or shortest chord passing through a given point, is also considered. For the solution the curve named ophiuride is used.