Question

If $S = \left\{z \in \mathbb{C} : \frac{z - i}{z + 2i} \in \mathbb{R}\right\}$, then :

• A. S contains only one element
• B. S contains exactly two elements
• C. S is a straight line in the complex plane
• D. S is a circle in the complex plane

Hint:

If $\frac{z-z_1}{z-z_2}$ is real, $z$ lies on the line passing through $z_1$ and $z_2$. Here $z_1 = i$ and $z_2 = -2i$, which are both on the imaginary axis.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
If a ratio of two complex numbers $\frac{z-z_1}{z-z_2}$ is purely real, then the points $z, z_1,$ and $z_2$ are collinear.
Step 2: Detailed Explanation:
Let $z = x + iy$.
\[ \frac{x + i(y-1)}{x + i(y+2)} = \frac{[x + i(y-1)][x - i(y+2)]}{x^2 + (y+2)^2} \]
For the expression to be real, the imaginary part of the numerator must be zero.
\[ \text{Im}([x + i(y-1)][x - i(y+2)]) = 0 \]
\[ -x(y+2) + x(y-1) = 0 \]
\[ -xy - 2x + xy - x = 0 \]
\[ -3x = 0 \implies x = 0 \]
$x=0$ represents the y-axis (imaginary axis) in the Argand plane, which is a straight line.
Step 3: Final Answer:
S is a straight line.