A characterization of the focals of hyperbolas (Q2337370)

The chord \(AB\) of a hyperbola revolve about an arbitrary point \(P\). The tangents to the hyperbola at the end points \(A\) and \(B\) intersect the asymptotes at the points \(A_1\) and \(A_2\), respectively \(B_1\) and \(B_2\). The author proves that the lines \(A_1 B_1\), \(AB\) and \(A_2B_2\) are parallel and \(AB\) is the middle-parallel of \(A_1B_1\) and \(A_2B_2\). The distance between the parallel \(AB\) and \(A_1B_1\) is variable, but when \(P\) is a focal point, this distance is independent of the direction of the chord \(AB\) and equal to \(b\), the conjugate axis of hyperbola.