Problem

Figure 5 shows the constraints of a linear programming problem in $x$ and $y$, where $R$ is the feasible region. The equations of two of the lines and the three intersection points, $A$, $B$ and $C$, are shown. The coordinates of $C$ are $\left(\dfrac{35}{4}, \dfrac{15}{4}\right)$.

The objective function is $P = x + 3y$.

When the objective is to maximise $x + 3y$, the value of $P$ is 24.

When the objective is to minimise $x + 3y$, the value of $P$ is 10.

(a) (i) Find the coordinates of $A$ and $B$.

(ii) Determine the inequalities that define $R$.

An additional constraint, $y \geq kx$, where $k$ is a positive constant, is added to the linear programming problem.

(b) Determine the greatest value of $k$ for which this additional constraint does not affect the feasible region.