The coordinates of the points P, Q and R are given by (-2, 1), (1, 4) and (4, k) respectively. Find the value(s) of k if
(a) P, Q and R lie on the same line (clue: gradient);
(b) PQ=QR (clue: distance).

Prove (a) cosh2(x) − sinh2(x) = 1 and (b) 1 − tanh 2(x) = sech 2(x). solution (a) cosh2(x) − sinh2(x) = ex + e−x 2 2 − 2 = e2x + 2 + e−2x 4 − = 4 = . (b) we start with the identity proved in part (a): cosh2(x) − sinh2(x) = 1. if we divide both sides by cosh2(x), we get 1 − sinh2(x) cosh2(x) = 1 or 1 − tanh 2(x) = .