Question

If $S = \\{z \in C : |z – i| = |z + i| = |z–1|\\}$, then, n(S) is:

• A. 1
• B. 0
• C. 3
• D. 4

Answer 1

Answer: (A)

1

Answer 2

Step 1. Interpret the Condition $|z - i| = |z + i| = |z - 1|$: This condition implies that $z$ is equidistant from the points $(0, 1)$, $(0, -1)$, and $(1, 0)$.

Step 2. Geometric Interpretation: The points $(0, 1)$, $(0, -1)$, and $(1, 0)$ form the vertices of an isosceles right triangle in the complex plane. The circumcenter of this triangle (the unique point equidistant from all three vertices) is the only point that satisfies the condition.

Step 3. Finding the Circumcenter: The circumcenter of a triangle with vertices $(0, 1)$, $(0, -1)$, and $(1, 0)$ lies at the origin $(0, 0)$.
Therefore, $z = 0$ is the only solution, so $n(S) = 1$.