2023 SAJC P1 Q6

The curve \({{C}_{1}}\) has equation \(y=\frac{x\left( x-a \right)}{x+a}\) and \(a\) is a positive real constant.

1. Find algebraically, in terms of \(a\), the range of values that \(y\) can take.[3]

It is now given that \(a=1\) and that curve \({{C}_{2}}\) has equation \(y=-\frac{5}{2}+\frac{10}{x+3}\).

2. Find the exact \(x\)-coordinates of the intersection points of \({{C}_{1}}\) and \({{C}_{2}}\).[2]
3. Sketch \({{C}_{1}}\) and \({{C}_{2}}\) on the same graph, giving the coordinates of any points where \({{C}_{1}}\) and \({{C}_{2}}\) intersect and the equations of any asymptotes.[4]
4. Find the exact area of the region bounded by the two curves \({{C}_{1}}\) and \({{C}_{2}}\), simplifying your answer in the form \(\left( a\ln 2+b\ln 5+c \right)\) units\(^{2}\), where \(a\), \(b\), \(c\) are constants to be determined.[3]