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Generalized “focus-directrix” property of conics. — Википедия Материал из Википедии — свободной энциклопедии

Generalized “focus-directrix” property of conics.
Generalized “focus-directrix” property. (Cf. [3, Theorem 1]) See
Figure 8. Let γ be a noncircular conic. Let ω be an arbitrary doubly tangent
circle of γ. Let λ be the line passing through the tangency points of ω and γ. If
the center of ω lies on the major axis of γ, then for an arbitrary point P ∈ γ we
have t(P,ω) = ε, where ε is the eccentricity of γ. If the center of ω lies on the d(P,λ)
minor axis of γ, then for an arbitrary point P ∈ γ we have t(P,ω) = ε′, where
ε′ = √ ε . |1−ε2 |
Proof. It suffices to consider the following 2 cases.
Case 1. The center of ω lies on the major axis of the conic γ or γ is a hyperbola
and the center of ω lies on the minor axis. The proof is found in [3, Theorem 1]. Case 2. The conic γ is an ellipse and the center of ω lies on the minor axis. Denote by Π the plane containing the circle ω. Consider the sphere Σ such that ω is a great circle of Σ; see Figure 9. Consider a circle θ ⊂ Σ such that γ is the orthogonal projection of θ onto the plane Π. It is easy to see that the eccentricity ε of γ is equal to sin φ, where φ is the angle between Π and the plane containing θ. Denote by O the center of ω. Consider the point N on θ such that PN⊥Π.Note that|PN|= |ON|2−|OP|2=t(P,ω).Therefore,
t(P,ω)= |PN| =tanφ=√ ε =ε′. d(P,λ) d(P,λ) 1 − ε2
Generalized “focus-directrix” property is proved.