2020 NYJC CT2 P1 Q6

The curve \({{C}_{1}}\) has equation \({{(y-a)}^{2}}-{{(kx+k)}^{2}}={{k}^{2}}\) where \(a>1\) and \(k>0\).

1. State a sequence of transformations which transforms the graph of \({{y}^{2}}-{{x}^{2}}=1\) to the graph of \({{C}_{1}}\).[3]

For this question, your sketch should include the equations of any asymptotes and the coordinates of any turning points.

2. 1. Given that \({{C}_{1}}\) intersects the \(x\)- axis at only one point, sketch the curve \({{C}_{1}}\).[3]
   2. On the same diagram, sketch the curve \({{C}_{2}}\) with equation[3]
      \(y=\frac{1}{a+k\sqrt{1+{{\left( x+1 \right)}^{2}}}}\)