Question from Gareth, a student:
2. Consider the rectangular hyperbola xy = c^2 with parameterization (x, y) = (ct, c/t), and t 6= 0.
i. Derive (i.e. do not just quote) the equations of the tangent and normal at the point P with
(parametric) coordinates (cp, c/p).
ii. Consider the points P : (cp, c/p) and Q : (cq, c/q) on the hyperbola. Find the equation of the
straight line that joins the points P and Q.
iii. Consider a point R : (cr, c/r) also on the hyperbola, and suppose that in the triangle PRQ,
\PRQ = 90◦. Prove that the normal (to the hyperbola) at R is parallel to the line PQ.