Complex Numbers And Quadratic Equations

Step 1: Roots of unity definition.

The 12^th roots of unity are the complex solutions to .

The 30^th roots of unity are the complex solutions to .

Step 2: Common solutions.

A complex number is a common root of both equations if and only if it satisfies

Equivalently, is a root of unity whose order divides both 12 and 30.

Step 3: Use greatest common divisor.

The common roots are exactly the -th roots of unity. Hence there are 6 common roots.

We are given the equation: and need to find the value of .

Step 1: Compute the denominator Expanding the denominator: Using the distributive property: Since , we substitute:

Step 2: Compute the fraction Rewriting: Multiply numerator and denominator by the conjugate : Computing the denominator: Expanding the numerator: Since , we simplify: Rewriting: Separating real and imaginary parts:

Step 3: Compute

Final Answer:

Step 1: Understanding the Given Condition
We are given three complex numbers with unit modulus, i.e., Also, we are given the equation:

Step 2: Expanding the Modulus Expressions
Expanding the squared modulus terms: Using properties of conjugates, Since , we simplify:

Step 3: Evaluating the Summation
Rearranging terms: Thus,

We are given the complex numbers:

Step 1: Compute
Dividing by : Multiply the numerator and denominator by the conjugate of the denominator : The denominator simplifies as: Expanding the numerator: Dividing by 4:

Step 2: Compute }
Since has a positive real part and a positive imaginary part, raising it to any power will still lie in the first quadrant.

Step 3: Conclusion
Thus, the point lies in: