Find the equation of the tangent to the rectangular hyperbola \(xy=c^2\) at the point \(P(ct,c/t)\).

A well drawn diagram will help us a lot in this question!

What is the gradient at any point on the curve \(xy=c^2\)?

The tangent at \(P\) meets the axes of \(x\) and \(y\) at \(L\) and \(M\) respectively. \(O\) is the centre of the hyperbola and \(POQ\) is a diameter. The straight line \(MQ\) meets the axis of \(x\) in \(T\). Prove that

  1. the area of triangle \(MOL\) is \(2c^2\),
  2. the area of triangle \(QOT\) is \(c^2/3\).

Can we use the equation of the tangent at \(P\) to find \(L\) and \(M\)?

What are the coordinates of \(Q\) if \(POQ\) is a diameter?

Note that a diameter of a hyperbola is any straight line passing through its centre. We can assume here that \(Q\) is the point on the hyperbola opposite \(P\).

illustration of a diameter for a hyperbola