# Mock AIME I 2012 Problems/Problem 8

## Problem

Suppose that the complex number satisfies . If is the maximum possible value of , can be expressed in the form . Find .

## Solution 1

We begin by dividing both sides by to obtain . Now, consider that we may write with a positive real number so that and some real number. Then, Since we need , we must have , or equivalently, . By the quadratic equation, this has roots and to maximize , we take the larger root which is clearly maximized when is minimized. Since , the maximum value of will occur where , so the maximum value of occurs where and finally we find that the maximum value of is Taking the fourth power, the desired answer is .

## Solution 2

Let

.

Note that

.

To compute , connect to the origin and construct an altitude to the x-axis. Extend this line one unit above and connect that point to the origin to create .

Now, we use the Law of Cosines on the triangle with vertices at the origin, , and , giving

(or you could use the Pythagorean Theorem). Either way, we proceed from here as we did with Solution 1 to get .

-Solution by **thecmd999**