JEE Advanced · 2008 · 13 Apr 2008 · Paper 1 · Complex Numbers

From the previous question, $z\in A\cap B\cap C$ is a specific point (there's only one such point).

For any point $z$ on circle $B:|z-2-i|=3$, we can use the property that if $P$ and $Q$ are two points, then for any point $Z$ on a circle:

The points $(-1,1)$ and $(5,1)$ are at distance $6$ apart, and their midpoint is $(2,1)$, which is the center of circle $B$.

For any point $z$ on the circle with center $(2,1)$ and radius $3$:


Using the identity for a point on a circle: if $A$ and $B$ are endpoints of a diameter, then for any point $P$ on the circle:


Here, $(-1,1)$ and $(5,1)$ are endpoints of a diameter of the circle (both are distance $3$ from center $(2,1)$).

Therefore:


This is a constant value, so it equals exactly $36$, which lies in the range 35 and 39.

Answer: (C)