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Curve studied by de Longchamps in 1890, who named it. 
Cartesian equation: ,
or (b > 0). Cartesian parametrization: . Rational cubic with isolated point (O). Polar equation: (written when a = b). 
Given a parabola (P) (here with equation ) and a line (D) perpendicular to the axis of (P) (here, with equation x = a), the associated mixed cubic is the locus of the point M on a variable line (D) passing by O, cutting (P) in P and cutting (D) in Q such that . In other words, the mixed cubic is the cissoid of the parabola (P) and the straight line (D); it is a special case of Zahradnik cissoid.
The name mixed cubic comes from the fact that this curve has a linear asymptote () as well as a parabolic asymptote ().
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© Robert FERRÉOL, Jacques MANDONNET 2017