A shaded region on a complex plane is shown - HSC - SSCE Mathematics Extension 2 - Question 8 - 2023 - Paper 1

To find the condition that describes the shaded region on the complex plane, let’s first analyze the geometric representation of the complex numbers involved. The complex number $z$ can be expressed as $z = x + yi$ where $x$ and $y$ are real numbers.

We recall that the expression $|z - i|$ represents the distance from the point $z$ to the point $i$ (which is $(0, 1)$ on the complex plane). The term $|z - 1|$ indicates the distance from $z$ to the point $1$ (which is $(1, 0)$ on the complex plane).

The condition $|z - 1| < 2|z - i|$ implies that the distance from $z$ to the point $1$ is less than twice the distance from $z$ to point $i$.

Given the graphical representation provided in the question, we can observe that the shaded region cannot extend far from the vicinity of the point $(0, 1)$ while remaining close to the point $(1, 0)$. This relationship implies that for points in the shaded area, they are constrained to being nearer to $(0, 1)$ than to $(1, 0)$ with a certain factor.

Hence, the relation that best describes the region shaded on the complex plane is:

Answer: $|z - 1| < 2|z - i|$