jmc

Question

Consider the two functions f(x) = x^2 + 2bx + 1 and g(x) = 2a(x + b), where the variables 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 points (a,b) for which the graphs of y = f(x) and y = g(x) do not intersect (in the xy-plane). Find the area of S.

Accepted Answer

The graphs intersect when f(x) = g(x) has a real root, or x^2 + 2bx + 1 = 2a(x + b).This simplifies to x^2 + (2b - 2a) x + (1 - 2ab) = 0. Thus, we want this quadratic to have no real roots, which means its discriminant is negative: (2b - 2a)^2 - 4(1 - 2ab) < 0.This simplifies to a^2 + b^2 < 1. This is the interior of the circle centered at (0,0) with radius 1, so its area is .