Conjugate Diameters in Ellipse

The line joining the midpoints of two parallel chords in an ellipse passes through the center of the ellipse. Moreover, the tangents to the ellipse at the extremities of the chords all meet on the same straight line.

Any chord that passes through the center of an ellipse is call its diameter. It follows that the family of parallel chords define two diameters: one in the direction to which they are all parallel and the other the locus of their midpoints. Such two diameters are called conjugate. The term suggests some symmetry between the two directions, the symmetry that is not apparent from the construction.

 

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There is indeed a symmetry:

  1. The midpoints of the chords parallel to one of the conjugate diameters lie on the other.
  2. The tangents at the end points of the chords parallel to one of the conjugate diameters meet on the other.
  3. To paraphrase the previous statement: the poles of polars parallel to one of the conjugate diameters lies on the other
  4. And vice versa: the polars of the poles on one of the conjugate diameters are parallel to the other.

The statement is easily proved analytically if we start with the equation x²/a² + y²/b² = 1 and the associated parameterization: x = a cos(t), y = b sin(t).

Thus ellipse is a curve defined by the radius-vector

  r(t) = (a cos(t), b sin(t)).

For a fixed t, we are interested in two points, r(t ± v). We shall use the addition formulas for sine and cosine:

 r(t + v) = (a (cos(t)cos(v) - sin(t)sin(v)), b (sin(t)cos(v) + cos(t)sin(v))),
 r(t - v) = (a (cos(t)cos(v) + sin(t)sin(v)), b (sin(t)cos(v) - cos(t)sin(v))).

The slope of the difference, say, r(t + v) - r(t - v) is -b/a cot(t), independent of v, meaning that we thus produce a family of parallel chords. Their midpoints satisfy

  (r(t + v) + r(t - v)) / 2 = cos(v) (a cos(t), b sin(t))

which is a parameterization (with parameter cos(v)) of the chord with the slope of b/a tan(t). To summarize, the midpoints of the chords parallel to the direction with the slope -b/a cot(t) lie on the line with the slope b/a tan(t). Applying the formulas of sine and cosine of the complementary angles we see that starting with the chords with the latter slope we would have found their midpoints on a line with the former slope, thus justifying the symmetric terminology. The two directions are conjugate. Observe in passing that, although the conjugate diameters correspond to the complementary values of the parameter t, the product of the two slopes is -b²/a² which is -1 only for circles, i.e. when a = b, so that in general, the conjugate diameters are not perpendicular.

Making use of the parameterization r(t) = (a cos(t), b sin(t)) I tacitly assumed that the origin of the system of coordinates has been placed at the center of the ellipse. We now evaluate the distance from the center to the end points of the conjugate diameters, i.e., P = (a cos(t), b sin(t)) and, say, Q = (-a sin(t), b cos(t)):

 OP² + OQ²= (a² cos²(t) + b² sin²(t)) + (a² sin²(t) + b² cos²(t))
  = (a² cos²(t) + a² sin²(t)) + (b² sin²(t) + b² cos²(t))
  = a² + b²,

independent of t. This is known as the first theorem of Apollonius: for the conjugate (semi)diameters OP and OQ, OP² + OQ² = a² + b².

The area of the parallelogram formed by the tangents parallel to a pair of conjugate diameters can be computed via the determinant:

 
A = 4
a·cos(t) b·sin(t) 1
-a·sin(t) b·cos(t) 1
0 0 1
= 4ab.

This is known as the second theorem of Apollonius.

References

  1. G. Salmon, Treatise on Conic Sections, Chelsea Pub, 6e, 1960
  2. C. Zwikker, The Advanced Geometry og Plane Curves and Their Applications, Dover, 2005

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