From Foci to a Tangent in Ellipse: What Is It About?
A Mathematical Droodle

 

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Explanation

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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.)

 

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


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 (F2 in the applet) to two parallel tangents. In the notations employed in the applet, the statement then reads

  F1P1 · F2P2 = F2P2 · F2P' = const.

This fact is hinted at by drawing a chord in the circle drawn on P2P' as diameter and perpendicular to P2P'. 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:

  UF2² = F2P2 · F2P'.

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 P2P' is a chord in the major circle through F2. By the intersecting chords theorem theproduct P2F2 · F2P' is independent of the direction of the cord.

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

  1. V. Gutenmacher, N. Vasilyev, Lines and Curves: A Practical Geometry Handbook , Birkhauser; 1 edition (July 23, 2004)

Conic Sections > Ellipse

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