1983 AMC 12 Problem 22

Question

Consider the two functions

\begin{aligned} f(x)&=x^2+2bx+1,\\ g(x)&=2a(x+b), \end{aligned}

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

1

\pi

4

4\pi

infinite

Set

f(x)=g(x)

and require the resulting quadratic to have no real roots

Simplify the negative-discriminant condition in terms of

a

and

b

Intersections correspond to roots of

x^2+2(b-a)x+(1-2ab)=0.

There are no real roots exactly when its discriminant is negative:

\begin{aligned} &4(b-a)^2-4(1-2ab)\\ &=4(a^2+b^2-1)\\ &\lt0. \end{aligned}

Thus

S

is the interior of the unit circle in the

ab

-plane, with area

\pi.

Therefore, the correct answer is B.

Problem 22 in other years