Problem:
Consider the two functions f(x)=x2+2bx+1 and g(x)=2a(x+b), where the variable x and the constants a and b are real numbers. Each such pair of constants a and b may be considered as a point (a,b) in an ab-plane. Let S be the set of such points (a,b) for which the graphs of y=f(x) and y=g(x) do not intersect (in the xy-plane). The area of S is
Answer Choices:
A. 1
B. Ο
C. 4
D. 4Ο
E. infinite
Solution:
We must describe geometrically those (a,b) for which the equation
x2+2bx+1=2a(x+b)
or equivalently, the equation
x2+2(bβa)x+(1β2ab)=0
has no real root for x. Since a quadratic equation Ax2+Bx+C=0 has no real root if and only if its discriminant B2β4AC is negative, S is the set of (a,b) for which
[2(bβa)]2β4(1β2ab)<04(a2β2ab+b2)β4+8ab<04a2+4b2<4a2+b2<1β
Thus S is the unit circle (without boundary) and the area is Ο.