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Let S be any curve and let A be a fixed point. If a straight line is drawn through A to meet S at Q, and if P and P' are points on this line such that 

P'Q = QP = k      (a constant)

the locus of P and P' is called a conchoid of S with respect to A.

The Conchoid of Nicomedes

Other Conchoids

The conchoid of a circle of radius r with respect to a point within it.
A conchoid of a parabola with respect to the focus
A conchoid of an ellipse or hyperbola with respect to a focus.