NJC Curve Sketching Tutorial Q2

Given that \(a\) is a positive constant. A curve \({{C}_{1}}\) has parametric equations

\(x=\frac{a}{t},y=1+t\).

Sketch \({{C}_{1}}\), labelling the coordinates of the point(s) where the curve crosses the \(x\)- and \(y\)-axes, and the equations of the asymptote(s) in terms of \(a\), if any.

Another curve \({{C}_{2}}\) has equation\(y=\sqrt{1+\frac{{{x}^{2}}}{{{a}^{2}}}}\).

1. Show algebraically that the \(y\)-coordinates of the point(s) of intersection of \({{C}_{1}}\) and \({{C}_{2}}\) satisfies the equation \({{\left( y-1 \right)}^{2}}\left( {{y}^{2}}-1 \right)-1=0\).
2. Sketch \({{C}_{2}}\) on the same diagram as \({{C}_{1}}\), labelling the coordinates of the point(s) where the curve crosses the \(x\)- and \(y\)-axes, and the equations of the asymptotes in terms of \(a\), if any.

Find the coordinates of the point(s) of intersections of \({{C}_{1}}\) and \({{C}_{2}}\) and label the coordinates in this diagram, leaving the answers correct to 3 significant figures, in terms of \(a\).