If z = x + yi is a complex number such that |z| = 10, what is the equation of th

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Question: If z = x + yi is a complex number such that |z| = 10, what is the equation of the circle in the complex plane?

Options:

Correct Answer: x^2 + y^2 = 100

Solution:

The equation of the circle with radius 10 is x^2 + y^2 = 10^2 = 100.

If z = x + yi is a complex number such that |z| = 10, what is the equation of the circle in the complex plane?

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.

Complex Numbers – Understanding the representation of complex numbers in the form z = x + yi, where x is the real part and y is the imaginary part.
Magnitude of Complex Numbers – The magnitude |z| of a complex number is calculated as √(x^2 + y^2), which relates to the distance from the origin in the complex plane.
Equations of Circles – The standard equation of a circle in the Cartesian plane is x^2 + y^2 = r^2, where r is the radius.