M241 - y^2=x^3 + ax and y=x+1
With the two equations, depending on what a is, you either have 1 intersecting point or 3. When a > -0.61(around), you have one intersection point and when a </= -.61 you have 3 intersection points. The reason for this is because when you have a </= -0.61, 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 = -0.61, 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 </= -0.61, 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. You will always have a line and a circle, where the circle intersects at two points, so therefore there will not be only 2 intersecting points, due to the equation shown above and the graph shown below.