2023 NYJC P1 Q8

The curve \(C\) is a circle with equation \({{x}^{2}}+{{y}^{2}}-8x-6y-20=0\).

1. \(C\) crosses the positive \(x\)-axis at point \(A\). Find the equation of \({{l}_{1}}\), the tangent to the circle at point \(A\).[4]
2. \(C\) crosses the positive \(y\)-axis at point \(B\). Find the gradient of \({{l}_{2}}\), the tangent to the circle at point \(B\). Hence, find the acute angle between the \({{l}_{1}}\) and \({{l}_{2}}\).[4]
3. The minor segment bounded by \(C\) and the y-axis is defined as \(R\). Find the volume generated when \(R\) is rotated four right angles about the \(y\)-axis, leaving your answer to \(1\) decimal place.[3]