Question

Sol. [Key Concept: A point lies in the angle (between two lines) that contains the origin iff it has the same sign as the origin in both line expressions.]

Intersection of $x = (y - 1)^{2}$ with lines are

$$\begin{matrix} & \begin{matrix} (y - 1)^{2} & \ = \frac{y - 1}{3} \Rightarrow y = 1,y = \frac{4}{3} \\ \Rightarrow \ & P \equiv \left( \frac{1}{9},\frac{4}{3} \right) \\ & \ (y - 1)^{2} = 5 - 2y \Rightarrow y^{2} = 4 \Rightarrow y = \pm 2 \\ & Q \equiv (1,2),R \equiv (9, - 2) \\ \Rightarrow \ & a + 1 \in ( - 2,1) \cup \left( \frac{4}{3},2 \right); \\ & a \in ( - 3,0) \cup \left( \frac{1}{3},1 \right) \end{matrix} \end{matrix}$$