Condition for Conjugate Diameters

Two diameters y = m₁x and y = m₂x of the ellipse x²/a² + y²/b² = 1 are conjugate iff m₁m₂ = −b²/a². Each bisects chords parallel to the other. For a circle, conjugate diameters are perpendicular (b=a gives m₁m₂=−1).

A diameter of an ellipse is a chord passing through the centre. For the ellipse $x^2/a^2 + y^2/b^2 = 1$, every diameter has the form $y = mx$.

Definition: Two diameters $y = m_1x$ and $y = m_2x$ are conjugate if the diameter $y = m_1x$ bisects all chords parallel to $y = m_2x$, and vice versa.

Derivation: From the chord-with-given-midpoint formula, the chord of $x^2/a^2 + y^2/b^2 = 1$ with midpoint $(x_1, y_1)$ has slope $-b^2x_1/(a^2y_1)$.

For the midpoint to lie on $y = m_1x$: $y_1 = m_1x_1$, so midpoint slope $= -b^2x_1/(a^2 \cdot m_1x_1) = -b^2/(a^2m_1)$.

For the chord to be parallel to $y = m_2x$: slope of chord $= m_2$.

$$m_2 = -\frac{b^2}{a^2m_1} \implies m_1m_2 = -\frac{b^2}{a^2}$$

Properties of conjugate diameters:

• Every pair of axes (major and minor) is a pair of conjugate diameters (with $m_1 \to \infty$, $m_2 = 0$)
• The parametric points $(a\cos\alpha, b\sin\alpha)$ and $(a\cos\beta, b\sin\beta)$ are endpoints of conjugate diameters iff $\alpha - \beta = \pm 90°$ (eccentric angles differ by 90°)
• Sum of squares of conjugate semi-diameters: $r_1^2 + r_2^2 = a^2 + b^2$ (constant)
• Product of areas of parallelograms formed by conjugate diameters: constant $= 4ab$

For a circle ($a = b$): $m_1m_2 = -1$ — conjugate diameters are perpendicular (ordinary conjugate directions).