Problem:
Let be a real number such that the system
has exactly one complex solution . The sum of all possible values of can be written as , where and are relatively prime positive integers. Find . Here .
Solution:
The first equation describes the set of points whose distance from in the complex plane is . This can be represented as a circle with center and radius .
The second equation describes the set of points equidistant from the points and . This is the perpendicular bisector of the line segment joining the two points. Thus the second equation is a line passing through with slope .
For there to be exactly one complex solution, the line must be perfectly tangent to the circle. If a line with slope is tangent to the circle, the radius drawn to the point of tangency will have slope . Since the radius is , this gives two possible points for the point of tangency or .
For the first point, using the fact that the slope between and is ,
Similarly for the second point:
Their sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions