8 A particle P is projected from a fixed point O with initial velocity \(\mathbf { u i } + \mathrm { kuj }\), where k is a positive constant. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the horizontal and vertically upward directions respectively. \(P\) moves with constant gravitational acceleration of magnitude \(g\). At time \(t \geqslant 0\), particle \(P\) has position vector \(\mathbf { r }\) relative to \(O\).

1. Starting from an expression for \(\ddot { \mathbf { r } }\), use integration to derive the formula $$\mathbf { r } = u \mathbf { t } + \left( k u t - \frac { 1 } { 2 } g t ^ { 2 } \right) \mathbf { j } .$$ The position vector \(\mathbf { r }\) of P at time \(\mathrm { t } \geqslant 0\) can be expressed as \(\mathbf { r } = x \mathbf { i } + y \mathbf { j }\), where the axes Ox and Oy are horizontally and vertically upwards through O respectively. The axis Ox lies on horizontal ground.
2. Show that the path of P has cartesian equation $$g x ^ { 2 } - 2 k u ^ { 2 } x + 2 u ^ { 2 } y = 0 .$$
3. Hence find, in terms of \(\mathrm { g } , \mathrm { k }\) and u , the maximum height of P above the ground during its motion. The maximum height P reaches above the ground is equal to the distance OA , where A is the point where P first hits the ground.
4. Determine the value of k .