N2010 P1 Q11

A curve \(C\) has parametric equations \(x=t+\frac1t\), \(y=t-\frac1t\).

1. The point \(P\) on the curve has parameter \(p\). Show that the equation of the tangent at \(P\) is \((p^2+1)x-(p^2-1)y=4p\).[4]
2. The tangent at \(P\) meets the line \(y=x\) at the point \(A\) and the line \(y=-x\) at the point \(B\). Show that the area of triangle \(OAB\) is independent of \(p\), where \(O\) is the origin.[4]
3. Find a cartesian equation of \(C\). Sketch \(C\), giving the coordinates of any points where \(C\) crosses the \(x\)- and \(y\)-axes and the equations of any asymptotes.[4]