2024 ACJC H1 Q5

The diagram shows the curve \(C\) with equation \(y = k^2 - x^2\), where \(k\) is a positive constant.

The \(x\)-coordinate of \(A\) is positive and the \(y\)-coordinate is \(\frac{3}{4}k^2\).

1. Find the equation of the tangent to \(C\) at the point \(A\). Give your answer in the form \(y = mx + c\), where \(m\) and \(c\) are in terms of \(k\).[4]
2. Find the area bounded by the curve \(C\), the tangent to the curve at point \(A\) and the \(x\)-axis, giving your answer in the form \(\frac{p}{q}k^3\), where \(p\) and \(q\) are integers.[4]