1983 AMC 12 Problem 17

Question

The diagram to the right shows several numbers in the complex plane. The circle is the unit circle centered at the origin. One of these numbers is the reciprocal of

F.

Which one?

A

B

C

D

E

Answer: C

Concepts:complex number coordinate geometry transformation

Difficulty rating: 1900

Small hint:

For

z=re^{i\theta},

its reciprocal has modulus

\frac{1}{r}

Big hint:

The reciprocal reflects the argument across the real axis and moves inside the unit circle

Solution:

If

F=re^{i\theta},

then

\frac{1}{F}=r^{-1}e^{-i\theta}.

Because

F

is outside the unit circle in quadrant I, its reciprocal is inside the unit circle in quadrant IV, with the reflected argument. Only point

C

has those properties.

Therefore, the correct answer is C.