2019 NYJC P2 Q2

The curve \(C\) has parametric equations \(x=6{{t}^{2}}\), \(y=\frac{2t}{\sqrt{1-{{t}^{2}}}}\), \(0 < t < 1\).

1. A line is tangent to the curve \(C\) at point \(A\) and passes through the origin \(O\). Show that the line has equation \(y=\frac{2}{3}x\).[4]

The region \(R\) is bounded by the curve and the tangent line in (i).

2. Find the area of \(R\).[3]
3. Write down the Cartesian equation of the curve \(C\).[1]
4. Find the exact volume of the solid of revolution generated when \(R\) is rotated completely about the \(x\)-axis, giving your answer in the form \(\left( a\ln b-c \right)\pi \), where constants \(a\), \(b\), \(c\) are to be determined.[4]