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\(A\) and \(B\) are two points at opposite ends of a diameter of a rectangular hyperbola, and \(P\) is a point which moves on the hyperbola. Prove that \(\angle PBA - \angle PAB\) is constant as long as \(P\) remains on the same branch of the curve and does not pass through \(A\) or \(B\). Find the relations between the four possible values of \(\angle PBA - \angle PAB\). If \(A\) and \(B\) are two given points, prove that the locus of \(P\), such that \[ \angle PBA - \angle PAB = \alpha, \] where \(\alpha\) is a given constant angle and \(P\) remains on the same side of \(AB\), is part of a rectangular hyperbola, and find the position of the transverse axis in relation to \(AB\).

Find the condition that the four lines given by the equations \[ ax^2+2hxy+by^2=0, \quad a'x^2+2h'xy+b'y^2=0 \] may represent a harmonic pencil. \par Shew that the same condition would make the first pair of lines conjugate diameters of the conic \[ a'x^2+2h'xy+b'y^2=1. \] Explain the relation of these results in reference to the theory of pencils in involution.