The points $P$ and $Q,$ as shown in Figure $2,$ have coordinates $(-2,13)$ and $(4,-5)$ respectively.

The straight line $l$ passes through $P$ and $Q.$

$($a$)$ Find an equation for $l,$ writing your answer in the form $y=mx+c,$ where $m$ and $c$

are integers to be found.

The quadratic curve $C$ passes through $P$ and has a minimum point at $Q.$

$($b$)$ Find an equation for $C.$

The region $R,$ shown shaded in Figure $2,$ lies in the second quadrant and is bounded by

$C$ and $l$ only.

$($c$)$ Use inequalities to define region $R.$