Vertical and horizontal gaps to a parabola

The curve $C$ and the line $L$ have equations $y=x^2$ and $y=\frac12x-2$ respectively.

(i) The point $A$ on $C$ and the point $B$ on $L$ are such that they have the same $x$-coordinate. Find the coordinates of $A$ and $B$ that gives the shortest distance $AB$.3 marks

(ii) The point $P$ on $C$ and the point $Q$ on $L$ are such that they have the same $y$-coordinate. Find the coordinates of $P$ and $Q$ that gives the shortest distance $PQ$.3 marks

(iii) Find the exact area of the polygon formed by joining the points found in (i) and (ii).2 marks

(iv) A variable point on $C$ with coordinates $(s,s^2)$ starts from origin $O$ and moves along the curve with $s$ increasing at a rate of 2 units/s. Find the rate of change of the area bounded by the curve, the $y$-axis and the line $y=s^2$, at the instant when $s=\sqrt2$.4 marks

[2017/ACJC/Prelim/I/8]