Point A has coordinates (p,q) where p<0 and q>0. Another point B is defined by interchanging and changing the sign of one of A's coordinates, so that B(q,−p).
Answer the following questions without assigning specific numerical values to p and q.
(i) Determine the quadrant in which point A lies and justify your answer.
(ii) Show logically, using the sign of each coordinate, in which quadrant point B must lie. Explain every step of your reasoning.
(iii) If a third point C(−p,−q) is considered, compare the locations of A and C. State whether they lie in the same quadrant, opposite quadrants or on an axis, and justify your statement.
(iv) The points A and B are joined to form a line segment. Prove that this segment must necessarily cross one of the coordinate axes. Identify the axis and support your proof with algebraic sign analysis of the end-point coordinates.