2024 YIJC H1 Q3

The diagram shows a triangular plot of ground, \(ABC\), in which \(AB=6\) m, \(AC=a\) m and angle \(BAC\) is a right angle. A gardener considers using a rectangular part, \(APQR\), of the triangle, where \(P\), \(Q\) and \(R\) lie on \(AB\), \(BC\) and \(AC\) respectively, to grow vegetables.

1. Given that \(AR=x\) m, show that \(AP=6-\frac{6}{a}x\) m.[2]
2. Given that \(x\) can vary, find the largest possible area of the vegetable plot \(APQR\), giving your answer in terms of \(a\). Justify that this is a maximum value.[4]
3. Given instead that the vegetable plot \(APQR\) is a square, find \(x\) in terms of \(a\).[1]