§ 73. The first part of the right-hand theorem in § 64 may be stated thus: any two conjugate lines through a point P without a conic are harmonic conjugates with regard to the two tangents that may be drawn from P to the conic.
If we take instead of P the centre C of an hyperbola, then the conjugate lines become conjugate diameters, and the tangents asymptotes. Hence—
Any two conjugate diameters of an hyperbola are harmonic conjugates with regard to the asymptotes.
As the axes are conjugate diameters at right angles to one another, it follows (§ 23)—
The axes of an hyperbola bisect the angles between the asymptotes.
| Fig. 26. |
Let O be the centre of the hyperbola (fig. 26), t any secant which cuts the hyperbola in C, D and the asymptotes in E, F, then the line OM which bisects the chord CD is a diameter conjugate to the diameter OK which is parallel to the secant t, so that OK and OM are harmonic with regard to the asymptotes. The point M therefore bisects EF. But by construction M bisects CD. It follows that DF = EC, and ED = CF; or
On any secant of an hyperbola the segments between the curve and the asymptotes are equal.
If the chord is changed into a tangent, this gives—
The segment between the asymptotes on any tangent to an hyperbola is bisected by the point of contact.