Two diameters y = m1x and y = m2x are conjugate if m1*m2 = -$b^2$/$a^2$. Each conjugate diameter bisects all chords parallel to the other. The major and minor axes are the unique pair of conjugate diameters that are also perpendicular. For conjugate semi-diameters CP and CD at eccentric angles alpha and beta: alpha - beta = pi/2 and $CP^2$ + $CD^2$ = $a^2$ + $b^2$ (invariant). The parallelogram formed by tangents at the four endpoints of conjugate diameters has constant area 4ab. The area of the triangle OPD = ab/2 (also constant). These properties are elegant but relatively rare in JEE compared to tangent and focal problems.