Question

Let $z=x+iy$ be a complex number, where $i=\sqrt{-1}$ is the complex unit. Then $|z-1+i|=5$ is a circle with:

• A. centre at (-1,1) and radius 5
• B. centre at (1,1) and radius $\sqrt{5}$
• C. centre at (-1,-1) and radius $\sqrt{5}$
• D. centre at (1,1) and radius 25
• E. centre at (1,-1) and radius 5

Hint:

Always factor out a negative sign to identify the centre $z_0$ correctly.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The equation \( |z - z_0| = r \) represents a circle in the complex plane (also known as the Argand plane). Here, \( z_0 \) is the complex number representing the center of the circle, and \( r \) is the radius. The modulus \( |z - z_0| \) represents the distance between the points corresponding to \( z \) and \( z_0 \).
Step 2: Key Formula or Approach:
We need to rewrite the given equation \( |z - 1 + i| = 5 \) in the standard form \( |z - z_0| = r \).
To do this, we factor out a negative sign from the terms inside the modulus to isolate \( z \).
\[ |z - (1 - i)| = 5 \] Step 3: Detailed Explanation:
Comparing the equation \( |z - (1 - i)| = 5 \) with the standard form \( |z - z_0| = r \), we can identify the center and the radius.
The center is \( z_0 = 1 - i \). In the complex plane, a complex number \( a + bi \) corresponds to the Cartesian coordinate \( (a, b) \). Therefore, the center of the circle is at the point \( (1, -1) \).
The radius is \( r = 5 \).
Alternative Method (using x and y):
Substitute \( z = x + iy \) into the original equation:
\[ |(x + iy) - 1 + i| = 5 \] Group the real and imaginary parts inside the modulus:
\[ |(x - 1) + i(y + 1)| = 5 \] The modulus of a complex number \( a + bi \) is \( \sqrt{a^2 + b^2} \).
\[ \sqrt{(x - 1)^2 + (y + 1)^2} = 5 \] Square both sides to get the standard Cartesian equation of a circle:
\[ (x - 1)^2 + (y - (-1))^2 = 5^2 \] This is the equation of a circle with center \( (h, k) = (1, -1) \) and radius \( r = 5 \).
Step 4: Final Answer:
The equation represents a circle with its centre at (1,-1) and a radius of 5.