1983 AHSME Problems/Problem 17

Problem

The diagram above 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?

$\textbf{(A)} \ A \qquad \textbf{(B)} \ B \qquad \textbf{(C)} \ C \qquad \textbf{(D)} \ D \qquad \textbf{(E)} \ E$

Solution

Write $F$ as $a + bi$ , where we see from the diagram that $a, b > 0$ and $a^2+b^2>1$ ( $F$ is outside the unit circle). We have $\frac{1}{a+bi} = \frac{a-bi}{a^2+b^2} = \frac{a}{a^2+b^2} - \frac{b}{a^2+b^2}i$ , so, since $a, b > 0$ , the reciprocal of $F$ has a positive real part and negative imaginary part. Also, the reciprocal has magnitude equal to the reciprocal of $F$ 's magnitude (since $|a||b| = |ab|$ ); as $F$ 's magnitude is greater than $1$ , its reciprocal's magnitude will thus be between $0$ and $1$ , so its reciprocal will be inside the unit circle. Therefore, the only point shown which could be the reciprocal of $F$ is point $\boxed{\textbf{C}}$ .

Page

Toolbox

Search

1983 AHSME Problems/Problem 17

Problem

Solution