The fact that the points of intersection are in line with the centre of the ellipse is just a coincidence caused by the choice of parameters. The unit circle centred at the origin goes through the points , and these happen to be the ends of the minor axis of the ellipse.
But the equation of the limašon is wrong, and the graph of it looks squashed up. It should extend out much further to the right of the origin. The equation for inversion is that the point (x,y) goes to . If you substitute for x and for y in the equation of the ellipse, then you get That is the equation of the limašon.