What is the difference between the diameter and conjugate diameter with respect to ellipse?

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 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. read less