CEMC Galois, 2006

In the diagram, a line is drawn through the points $A(0,16)$ and $B(8,0)$. Point $P$ is chosen in the first quadrant on the line through $A$ and $B$. Points $C$ and $D$ are then chosen on the $x$-axis and $y$-axis, respectively, so that $PDOC$ is a rectangle.

(a) Determine the equation of the line through $A$ and $B$.

(b) Determine the coordinates of the point $P$ so that $PDOC$ is a square.

(c) Determine the coordinates of all points $P$ that can be chosen so that the area of rectangle $PDOC$ is 30.