Problem:
The parabola with equation y=x2β4 is rotated 60β counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has y-coordinate caβbββ, where a, b, and c are positive integers, and a and c are relatively prime. Find a+b+c.
Solution:
We can use complex numbers to rotate points by any number of degrees around the origin. In particular multiplying z by cisΞΈ rotates it ΞΈ degrees counter-clockwise. Consider the point in question (a,b). If we rotate (a,b)60β clockwise it should land on the parabola:
Factoring the first quadratic gives us 23β57ββ, the solution we are looking for, as the other quadratic's negative solution gives us a solution in the third quadrant instead of the fourth. 3+57+2β62β.