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 + X^{2} + C
and,
X^{2} + Y^{2} = 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 yaxis that is on the intersection of these two curves will be matched by one that is symmetrically opposite w/r/t the yaxis.
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 20081017 15:10:31 