Determine all possible value(s) of a real constant C such that the following system of equations has precisely one real solution in (X, Y).

2^|X| + |X| = Y + X² + C

and, X² + Y² = 1

Note: |X| denotes the absolute value of X.

If (X,Y) is a solution to this system of equations than (-X,Y) is also a solution, regardless of the value of C. This follows from the fact that |X| = |-X| and X² = (-X)²

So, for there to be exactly one real solution, X=-X=0, reducing the system to:

1+0=Y+0+C => Y+C=1

and Y² = 1

Y, then must either be 1 or -1 for the second equation to hold. If Y=1, then C=0 from the first equation; if Y=-1, then C=2 from the first equation.

The only values of C, then, where this system has a unique solution are C = 0, C=2 and the unique solutions are (0,1) and (0,-1) respectively.

Posted by Paul on 2008-10-17 15:05:55