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.

Written as:

y = 2^abs(x) + abs(x) - x^2 - c,

this is an even function; that is, if a point (x1, y1) is on the curve, then point (-x1,y1) is also on the curve.

Though not a function, the relation x^2 + y^2 = 1 also has the property that if a point (x1, y1) is on the curve, then point (-x1,y1) is also on the curve. Thus any point to the left or right of the y-axis that is on the intersection of these two curves will be matched by one that is symmetrically opposite w/r/t the y-axis.

So any solution in (x,y) must have x = 0.

With x = 0, the first equation becomes:

y = 1 - c

and the second becomes:

y^2 = 1

The latter implies y = +/- 1.

If y is to be 1, then c is 0 by the first equation.

If y is to be -1, then c is 2 by the first equation.

So c can only be zero or 2.

When c is zero, the real solution is (0,1).

However, when c is 2, there are two other solutions besides the one at (0,-1). They are (1,0) and (-1,0).

So zero is the only value for c that results in a single solution.

Posted by Charlie on 2008-10-17 15:10:31