y=x+1 and y^2=x^3+ax
With the two equations, depending on what a is, you either have 1 intersecting point or 3. When a > -0.7, you have one intersection point and when a </= -.07 you have 3 intersection points. The reason for this is because when you have a </= -0.7, the solutions for x become complex, creating intersections with the circle as well as the cubic line. When you have y=x+1 and y^2=x^3 + ax, using substitution, it gives you (x+1)^2 = x^3 + ax, using addition, subtraction and multiplication the end result equals. 0 = x^3 - x^2 - (2+a)x - 1. So for example when you have a = -0.7, you have complex solutions x = -.445911 (+/-) 0.698807i, and a real solution 1.93741, which is why one point will always intersect, yet for a </= -.07, you have two more intersections by complex. Which shows why at that point they intersection with each other among the extra circle and when they don't, due to complex solutions.