Locus of |z − (1 + i)| = 1
What “locus” means: a locus is the set of all points that satisfy a given condition.
We start with the given condition:
|z − (1 + i)| = 1
The “| · |” symbol means the modulus, which is the length of a complex number. This represents its distance from the origin (0) in the Argand plane.
So, |z − (1 + i)| represents the distance from the point z to the fixed point (1 + i) in the complex plane.
The condition |z − (1 + i)| = 1 means all points z that are exactly 1 unit away from the point (1 + i).
To work with this algebraically, represent z using coordinates. Let z be:
z = x + i y
(where x is the real part and y is the imaginary part).
Now substitute this into the original condition:
|(x + i y) − (1 + i)| = 1
Subtract the real and imaginary parts separately:
(x + i y) − (1 + i) = (x − 1) + i (y − 1)
In this expression, the real part is (x − 1) and the imaginary part is (y − 1).
The modulus of a complex number a + i b is given by:
|a + i b| = √(a² + b²)
Apply this formula here with a = (x − 1) and b = (y − 1):
|(x − 1) + i (y − 1)| = √((x − 1)² + (y − 1)²)
The condition states that this modulus equals 1:
√((x − 1)² + (y − 1)²) = 1
Since both sides are non-negative, square both sides to eliminate the square root:
(x − 1)² + (y − 1)² = 1
This is the equation of a circle in the xy-plane with center at (1, 1) and radius 1.
Locus: The set of all points (x, y) satisfying (x − 1)² + (y − 1)² = 1 corresponds to all complex numbers z on this circle, where the distance from z to (1 + i) is 1.
The Argand plane is the complex plane, with the horizontal axis as the real part (Re) and the vertical axis as the imaginary part (Im).