For a unit circle with parametric equations
the negative pedal curve with respect to the pedal point is
Therefore if the point is inside the circle (), the negative pedal is an ellipse, if , it is a single point, if the point is outside the circle (), the negative pedal is a hyperbola.
More things to try:
Weisstein, Eric W. "Circle Negative Pedal Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CircleNegativePedalCurve.html