The Inverse of a Parabola is a Cardioid
In the figure below, the origin is the focus of the green parabola, and the blue unit circle () is shown in blue. The inverse of the parabola with respect to the unit circle is the red cardioid. As we can see, and are reciprocals. So, it follows that
If you select the "Tangent line and inverse" option, you will see the tangent line to the parabola and the inverse of the tangent line. This produces a circle that is tangent to the cardioid at A. If you select the circle option, you will see that the circle's center is on the cardioid's generating circle.