Can we find the triangle areas given by a tangent?

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

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 \(c^2/3\).