1991 AMC 12 Problem 18

Attempt Problem 18 of the 1991 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1991 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

18.

S

is the set of points

z

in the complex plane such that

(3+4i)z

is a real number, then

S

is a

right triangle

circle

hyperbola

line

parabola

Write

z=x+yi

and expand the product

Set the imaginary part of

(3+4i)(x+yi)

equal to zero

Writing

z=x+yi,

\begin{aligned} (3+4i)z &=(3x-4y)\\ &\quad +(4x+3y)i. \end{aligned}

This is real exactly when

4x+3y=0,

which is the equation of a line through the origin.

Thus the correct answer is D.

Problem 18 in other years