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. π
C. 4
D. 4Ï€
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(b−a)x+(1−2ab)=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 B2−4AC is negative, S is the set of (a,b) for which
[2(b−a)]2−4(1−2ab)<04(a2−2ab+b2)−4+8ab<04a2+4b2<4a2+b2<1​
Thus S is the unit circle (without boundary) and the area is π.