Can we show these tangents to this hyperbola form a square?

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Sketch on the same axes the two curves \(C_1\) and \(C_2\), given by \[\begin{align*} C_1&: xy &= 1, \\ C_2&: x^2-y^2 &= 2. \end{align*}\]

The curves intersect at \(P\) and \(Q\). Given that the coordinates of \(P\) are \((a,b)\) (which you need not evaluate), write down the coordinates of \(Q\) in terms of \(a\) and \(b\).

The tangent to \(C_1\) through \(P\) meets the tangent to \(C_2\) through \(Q\) at the point \(M\), and the tangent to \(C_2\) through \(P\) meets the tangent to \(C_1\) through \(Q\) at \(N\). Show that the coordinates of \(M\) are \((-b,a)\) and write down the coordinates of \(N\).

Show that \(PMQN\) is square.