If z = x + yi is a complex number such that |z| = 10, what is the equation of the circle in the complex plane?
Correct Answer: x^2 + y^2 = 100
- Step 1: Understand that a complex number z can be written as z = x + yi, where x is the real part and y is the imaginary part.
- Step 2: The magnitude (or absolute value) of the complex number z is given by |z| = sqrt(x^2 + y^2).
- Step 3: We are given that |z| = 10, which means sqrt(x^2 + y^2) = 10.
- Step 4: To eliminate the square root, we square both sides of the equation: (sqrt(x^2 + y^2))^2 = 10^2.
- Step 5: This simplifies to x^2 + y^2 = 100.
- Step 6: The equation x^2 + y^2 = 100 represents a circle in the complex plane with a radius of 10.
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