2025 CJC Promo Q5

The diagram below shows a parabola which cuts the \(x\)-axis at \((-2a,0)\), \((2a,0)\) and has a maximum point at \((0,4a^2)\), where \(a\) is a positive constant. A triangle \(OPQ\) is inscribed in the parabola such that the points \(P\) and \(Q\) lie symmetrically on the parabola at \(x=\pm k\) (where \(0<k<2a\)) and \(O\) is the origin.

1. Write down an equation of the parabola in terms of \(a\).[1]
2. By considering the coordinates of \(P\), show that \(A\), the area of triangle \(OPQ\), can be written as[2]
   \(A=k\left(4a^2-k^2\right).\)
3. Hence using differentiation, find the maximum value of \(A\) in terms of \(a\) as \(k\) varies.[5]