From Foci to a Tangent in Ellipse: What Is It About?
A Mathematical Droodle
What if applet does not run? |
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Copyright © 1996-2018 Alexander Bogomolny
From Foci to a Tangent in Ellipse
The applet attempts to illustrate the following fact [Gutenmacher and Vasilyev, 6.12(a)]:
The product of distances from the foci of an ellipse to any tangent is a constant (not depending on the particular tangent.) |
What if applet does not run? |
The proof depends on a property of the perpendiculars from the foci to tangents: the feet of the perpendiculars lie on a circle, the major circle of the ellipse.
The perpendiculars from the two foci to a tangent can be represented by the perpendiculars from a single foci (F_{2} in the applet) to two parallel tangents. In the notations employed in the applet, the statement then reads
F_{1}P_{1} · F_{2}P_{2} = F_{2}P_{2} · F_{2}P' = const. |
This fact is hinted at by drawing a chord in the circle drawn on P_{2}P' as diameter and perpendicular to P_{2}P'. Such a chord is twice the altitude to the hypotenuse in a right triangle in which the right angle is subtended by the diameter. The altitude is known to be the geometric mean of the pieces of the hypotenuse:
UF_{2}² = F_{2}P_{2} · F_{2}P'. |
Thus the altitude and with it the chord UV are expected to be constant, independent of the chosen tangent.
The proof of the statement, however, needs only the intersecting chords theorem and the existence of the major circle. The latter is unique for a given ellipse. Segment P_{2}P' is a chord in the major circle through F_{2}. By the intersecting chords theorem theproduct
It is easy to find the exact value of the constant product. Suffice to this end to choose the tangents parallel to the major axis. The product is then the square of the minor semiaxis and remains the same for all positions of the tangents.
References
- V. Gutenmacher, N. Vasilyev, Lines and Curves: A Practical Geometry Handbook , Birkhauser; 1 edition (July 23, 2004)
|Activities| |Contact| |Front page| |Contents| |Geometry| |Eye opener|
Copyright © 1996-2018 Alexander Bogomolny