1983 AHSME Problems/Problem 22

Problem

Consider the two functions $f(x) = x^2+2bx+1$ 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

$\textbf{(A)} \ 1 \qquad \textbf{(B)} \ \pi \qquad \textbf{(C)} \ 4 \qquad \textbf{(D)} \ 4 \pi \qquad \textbf{(E)} \ \text{infinite}$

Solution

We must describe geometrically those $(a,b)$ for which the equation $x^2+2bx+1=2a(x+b)$ , i.e. $x^2+2(b-a)x+(1-2ab)=0$ , has no solutions (equivalent to the graphs not intersecting). By considering the discriminant of this quadratic equation, there are no solutions if and only if $\left(2(b-a)\right)^2 - 4(1)(1-2ab) < 0 \Rightarrow (b-a)^2 < 1 - 2ab \Rightarrow a^2 - 2ab + b^2 < 1 - 2ab \Rightarrow a^2 + b^2 < 1$ . Thus $S$ is the unit circle (without its boundary, due to the inequality sign being $<$ rather than $\leq$ , but this makes no difference to the area), whose area is $\pi (1^2) = \pi$ , so the answer is $\boxed{\textbf{(B)} \ \pi}$ .

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1983 AHSME Problems/Problem 22

Problem

Solution