3 (In this question, use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).)
A golf ball is hit from a point \(O\) on a horizontal golf course with a velocity of \(40 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\theta\). The golf ball travels in a vertical plane through \(O\). During its flight, the horizontal and upward vertical distances of the golf ball from \(O\) are \(x\) and \(y\) metres respectively.

1. Show that the equation of the trajectory of the golf ball during its flight is given by $$x ^ { 2 } \tan ^ { 2 } \theta - 320 x \tan \theta + \left( x ^ { 2 } + 320 y \right) = 0$$
2. 1. The golf ball hits the top of a tree, which has a vertical height of 8 m and is at a horizontal distance of 150 m from \(O\). Find the two possible values of \(\theta\).
   2. Which value of \(\theta\) gives the shortest possible time for the golf ball to travel from \(O\) to the top of the tree? Give a reason for your choice of \(\theta\).