Question

If \( z \) is a complex number such that \( \frac{z - i}{z - 1} \) is purely imaginary, then the minimum value of \( |z - (3 + 3i)| \) is :

• A. \( 2\sqrt{2} - 1 \)
• B. \( 2\sqrt{2} \)
• C. \( 3\sqrt{2} \)
• D. \( 6\sqrt{2} \)

Hint:

In complex geometry, \( \text{Re}(\frac{z-z_1}{z-z_2}) = 0 \) always represents a circle with diameter endpoints \( z_1 \) and \( z_2 \).
Here \( z_1 = i \) and \( z_2 = 1 \). Midpoint is \( (1/2, 1/2) \) and radius is half the distance between \( (1, 0) \) and \( (0, 1) \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A complex number \( w \) is purely imaginary if \( \text{Re}(w) = 0 \).
Geometrically, if \( \frac{z - z_1}{z - z_2} \) is purely imaginary, the locus of \( z \) is a circle where the segment joining \( z_1 \) and \( z_2 \) is the diameter (excluding endpoints).
Step 2: Key Formula or Approach:
Let \( z = x + iy \).
Substitute into the expression \( \frac{z - i}{z - 1} \) and set the real part to zero to find the equation of the circle.
The minimum value of \( |z - z_0| \) is the distance from \( z_0 \) to the center of the circle minus the radius.
Step 3: Detailed Explanation:
Let \( z = x + iy \). The given expression is:
\[ w = \frac{x + i(y - 1)}{(x - 1) + iy} \]
Multiply the numerator and denominator by the conjugate of the denominator:
\[ w = \frac{[x + i(y - 1)] \cdot [(x - 1) - iy]}{(x - 1)^2 + y^2} \]
For \( w \) to be purely imaginary, \( \text{Re}(w) = 0 \):
\[ x(x - 1) + y(y - 1) = 0 \implies x^2 - x + y^2 - y = 0 \]
Completing the square:
\[ (x - \frac{1}{2})^2 + (y - \frac{1}{2})^2 = \frac{1}{2} \]
This is a circle with center \( C(\frac{1}{2}, \frac{1}{2}) \) and radius \( R = \frac{1}{\sqrt{2}} \).
We need the minimum value of distance from \( z \) to \( P(3, 3) \).
Distance \( CP = \sqrt{(3 - \frac{1}{2})^2 + (3 - \frac{1}{2})^2} = \sqrt{(\frac{5}{2})^2 + (\frac{5}{2})^2} = \sqrt{\frac{25}{4} + \frac{25}{4}} = \sqrt{\frac{50}{4}} = \frac{5}{\sqrt{2}} \).
Minimum distance \( = CP - R = \frac{5}{\sqrt{2}} - \frac{1}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \).
Step 4: Final Answer:
The minimum value is \( 2\sqrt{2} \).