(a)
(i) \( y = \frac{2}{3\sqrt{3}} x \)
(ii) \( P = \left( 3\sqrt{3}, 2 \right) \)

Method 1:
Parametric derivatives: \( \frac{dx}{dt} = 3t^2 \), \( \frac{dy}{dt} = 2t \)
Slope: \( \frac{dy}{dx} = \frac{2t}{3t^2} = \frac{2}{3t} \)
Tangent at \( P(p^3, p^2 – 1) \): \( y – (p^2 – 1) = \frac{2}{3p} (x – p^3) \)
Through origin: \( -(p^2 – 1) = \frac{2}{3p} \times (-p^3) \)
Solve: \( p^2 – 1 = \frac{2p^2}{3} \implies \frac{p^2}{3} = 1 \implies p = \sqrt{3} \)
Coordinates: \( x = (\sqrt{3})^3 = 3\sqrt{3} \), \( y = (\sqrt{3})^2 – 1 = 2 \)
Slope: \( m = \frac{2}{3 \times \sqrt{3}} = \frac{2}{3\sqrt{3}} \)
Line: \( y = \frac{2}{3\sqrt{3}} x \)

Method 2:
Line: \( y = mx \)
Intersection: \( t^2 – 1 = m t^3 \implies m t^3 – t^2 + 1 = 0 \)
Stationary points: \( 3m t^2 – 2t = 0 \implies t = \frac{2}{3m} \)
Substitute: \( m \left( \frac{2}{3m} \right)^3 – \left( \frac{2}{3m} \right)^2 + 1 = 0 \)
Solve: \( \frac{8}{27m^2} – \frac{4}{9m^2} + 1 = 0 \implies m^2 = \frac{4}{27} \implies m = \frac{2}{3\sqrt{3}} \)
Find \( t \): \( t = \frac{2}{3 \times \frac{2}{3\sqrt{3}}} = \sqrt{3} \)
Coordinates: \( x = (\sqrt{3})^3 = 3\sqrt{3} \), \( y = (\sqrt{3})^2 – 1 = 2 \)
Line: \( y = \frac{2}{3\sqrt{3}} x \)

Result: \( y = \frac{2}{3\sqrt{3}} x \), \( P = \left( 3\sqrt{3}, 2 \right) \) [8]

(b)
\( Q = \left( -\frac{3\sqrt{3}}{8}, -\frac{1}{4} \right) \)

Intersection: \( t^2 – 1 = \frac{2}{3\sqrt{3}} t^3 \)
Simplify: \( 2 t^3 – 3\sqrt{3} t^2 + 3\sqrt{3} = 0 \)
Double root: \( t = \sqrt{3} \)
Sum of roots: \( \sqrt{3} + \sqrt{3} + t_3 = \frac{3\sqrt{3}}{2} \implies t_3 = -\frac{\sqrt{3}}{2} \)
Coordinates: \( x = \left( -\frac{\sqrt{3}}{2} \right)^3 = -\frac{3\sqrt{3}}{8} \), \( y = \left( -\frac{\sqrt{3}}{2} \right)^2 – 1 = -\frac{1}{4} \)

Result: \( Q = \left( -\frac{3\sqrt{3}}{8}, -\frac{1}{4} \right) \) [5]