2022 YIJC Promo Q7

The curve \({{C}_{1}}\) has equation \(y=\frac{{{x}^{2}}-5}{x+3}\).

The curve \({{C}_{2}}\) has equation \(\frac{{{\left( x+5 \right)}^{2}}}{4}+\frac{{{\left( y+6 \right)}^{2}}}{{{b}^{2}}}=1\), where \(b>0\).

1. For \({{C}_{1}}\), find algebraically the range of values that \(y\) can take.[3]
2. Given that \(b=2\), sketch \({{C}_{1}}\) and \({{C}_{2}}\) on the same graph, giving the exact coordinates of any points where \({{C}_{1}}\) or \({{C}_{2}}\) crosses the \(x\)-axis and the equations of any asymptotes.[5]
3. 1. State the range of values of \(m\) such that \(y+6=m\left( x+3 \right)\) does not intersect \({{C}_{1}}\).[1]
   2. State the range of values of \(b\) such that there are two distinct points of intersection between \({{C}_{1}}\) and \({{C}_{2}}\).[1]