The position vector of a particle at time seconds is given by

, where .

1. Write in the form , where .  (1 mark)

2. Show that the Cartesian equation of the path of the particle is   (2 marks)

3. The particle is at point when and at point when , where is a positive real constant.
4. If the distance travelled along the curve from to is , find .   (1 mark)

4. Find all values of for which the position vector of the particle, , is perpendicular to its velocity vector, .   (2 marks)