HyperbolaAsEnvelopeII

Hyperbola as envelope II

Point E glides on circle B(BE). Point H is taken on AE (A fixed, outside the circle) such that AH/AE = k (constant).
Line HU is taken to be orthogonal at H to AE.
Line HU envelopes a hyperbola.
This hyperbola has asymptotes forming an angle equal to the angle of tangents to B(BE) from A.
The auxiliary circle of the hyperbola is homothetic to the circle B(BE) w.r. to A and in ratio k.
The hyperbolas resulting for various k are all homothetic to each other.
The contact point U of line HU with the hyperbola is the projection on HU of T, which is the harmonic fourth of K(A,S), where K is the intersection of HU with the line of centers AB.
These hyperbolas are homothetic to the one discussed in HyperbolaAsEnvelope.html , hence the results can be deduced from this but are also immediate consequences of the properties of hyperbolas studied in Hyperbola.html .

Hyperbola as envelope II

See Also