Where do two tangents cross, and where do two normals cross?

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The curve \(C\) has equation \(xy=\frac{1}{2}\). The tangents to the curve \(C\) at the distinct points \(P(p,\frac{1}{2p})\) and \(Q(q,\frac{1}{2q})\), where \(p\) and \(q\) are positive, intersect at \(T\) and the normals to \(C\) at these points intersect at \(N\). Show that \(T\) is the point \[\left(\frac{2pq}{p+q},\frac{1}{p+q}\right).\] In the case \(pq=\frac{1}{2}\), find the coordinates of \(N\). Show (in this case) that \(T\) and \(N\) lie on the line \(y=x\) and are such that the product of their distances from the origin is constant.