The tangents to an ellipse and a homofocal hyperbola at each one of their intersection points P are perpendicular. The circle through the point P and the common focal points F1, F2 passes also through the intersection points Q1, Q2 of the tangents with the minor axis.
The two conics, considering their geometric definition as loci having constant sum/difference of distances from the two foci (F1, F2 common to the two conics), have their tangents coincide with the exterior/interior bisector of angle F1PF2 (see Ellipse.html ).
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Homofocal ellipse and hyperbola ![[0_0]](Homofocal_files/Homofocal_0_0_0.png)
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![[1_1]](Homofocal_files/Homofocal_0_1_1.png)