2025 TGM P2 Q1

A curve is defined parametrically by the equations

\(x = \frac{4t}{1-t^2}, \ y = \frac{1}{1-t^2} \quad \text{where}\hspace{0.5em} t \in \mathbb{R}.\)

1. Show that the equation of the tangent to the curve at point \(P\left(2\frac{2}{3}, 1\frac{1}{3}\right)\) is \(5y = x + 4\).[4]
2. Determine if the tangent cuts the curve again.[3]