graphing |z-1|=2 , in the complex plane (related videos in desc. below)

We start with the equation |z - 1| = 2, where z is a complex number.
Let z = x + yi, where x and y are real numbers. Substituting this into the equation, we get:
|(x + yi) - 1| = 2
Simplifying the left-hand side, we have:
√((x - 1)² + y²) = 2
Squaring both sides of the equation, we get:
(x - 1)² + y² = 4
This is the equation of a circle with center (1,0) and radius 2. To see this, we can use the standard form equation for a circle:
(x - h)² + (y - k)² = r²
where (h,k) is the center of the circle and r is the radius. Comparing this with the equation we obtained, we see that h = 1, k = 0, and r = 2, which gives us the equation:
(x - 1)² + y² = 2²
To graph the circle on the x-y plane, we plot the center point (1,0) and then draw a circle with radius 2 around it. This gives us the graph of the equation |z - 1| = 2 in the Cartesian plane, which is a circle with center (1,0) and radius 2.
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Here's how we can graph the equation |z - 1| = 1, where z is a complex number:
Let z = x + yi, where x and y are real numbers. Substituting this into the equation, we get:
|(x + yi) - 1| = 1
Simplifying the left-hand side, we have:
√((x - 1)² + y²) = 1
Squaring both sides of the equation, we get:
(x - 1)² + y² = 1
This is the equation of a circle with center (1,0) and radius 1. To see this, we can use the standard form equation for a circle:
(x - h)² + (y - k)² = r²
where (h,k) is the center of the circle and r is the radius. Comparing this with the equation we obtained, we see that h = 1, k = 0, and r = 1, which gives us the equation:
(x - 1)² + y² = 1²
To graph the circle on the x-y plane, we plot the center point (1,0) and then draw a circle with radius 1 around it. This gives us the graph of the equation |z - 1| = 1 in the Cartesian plane, which is a circle with center (1,0) and radius 1.
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Here's how we can graph the equation |z + 1| = 2, where z is a complex number:
Let z = x + yi, where x and y are real numbers. Substituting this into the equation, we get:
|(x + yi) + 1| = 2
Simplifying the left-hand side, we have:
√((x + 1)² + y²) = 2
Squaring both sides of the equation, we get:
(x + 1)² + y² = 4
This is the equation of a circle with center (-1,0) and radius 2. To see this, we can use the standard form equation for a circle:
(x - h)² + (y - k)² = r²
where (h,k) is the center of the circle and r is the radius. Comparing this with the equation we obtained, we see that h = -1, k = 0, and r = 2, which gives us the equation:
(x + 1)² + y² = 2²
To graph the circle on the x-y plane, we plot the center point (-1,0) and then draw a circle with radius 2 around it. This gives us the graph of the equation |z + 1| = 2 in the Cartesian plane, which is a circle with center (-1,0) and radius 2.

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[Ydmaster]

You explained this better than my professor. Thank you. You enlightened me.

[reecehyrkas]

Thanks for fully explaining this. My book just gave the answer in an example, so I was doing some complicated ways of trying to think about it. This is much simpler than what I was trying to do.