Draw a circle of oscillation for a manipulable function of Cornu

I need to draw an interactive horn function like this:

Manipulate[
 ParametricPlot[{FresnelC[M*t]Fresnel[M*t]}, {t, 0, 1},
PlotRange -> {{0, 1}, {0, 1}}], {M, 0, 3}]

But I must also add an osculant circle to the parcel of land at its intersection where the tip of the parcel is such that when I manipulate the horn, the circle is drawn to illustrate the radius of curvature at that location.

I've tried several different methods to generate the circle, but none of them worked for me. Any help would be appreciated.