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.
|What if applet does not run?|
There is indeed a symmetry:
- The midpoints of the chords parallel to one of the conjugate diameters lie on the other.
- The tangents at the end points of the chords parallel to one of the conjugate diameters meet on the other.
- To paraphrase the previous statement: the poles of polars parallel to one of the conjugate diameters lies on the other
- 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
Thus ellipse is a curve defined by the radius-vector
|r(t) = (a cos(t), b sin(t)).|
|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
|(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
Making use of the parameterization
|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,
The area of the parallelogram formed by the tangents parallel to a pair of conjugate diameters can be computed via the determinant:
This is known as the second theorem of Apollonius.
- G. Salmon, Treatise on Conic Sections, Chelsea Pub, 6e, 1960
- C. Zwikker, The Advanced Geometry og Plane Curves and Their Applications, Dover, 2005