Quick questions on Curves and regions in the complex plane - TCE Mathematics Specialised (Tasmania)

The equation $|z - a| = r$ says the distance from $z$ to the fixed point $a$ is constant and equal to $r$. That is exactly the definition of a circle of radius $r$ centred at $a$. If $a = p + iq$, writing $z = x + iy$ gives

The equation $|z - a| = |z - b|$ says $z$ is equidistant from the two fixed points $a$ and $b$. The set of such points is the perpendicular bisector of the segment joining $a$ and $b$. The inequality $|z - a| < |z - b|$ is the half-plane of points closer to $a$.

The equation $\arg(z - a) = \theta$ fixes the direction from the point $a$ to $z$. The locus is a ray (half-line) starting at $a$, pointing at angle $\theta$ to the positive real axis. The starting point $a$ itself is excluded, because $\arg(0)$ is undefined. A condition like $\alpha \le \arg(z - a) \le \beta$ describes a wedge-shaped region between two rays.

A condition like $|z + 1| = \sqrt{2}\,|z - i|$, where the two distances are in a fixed ratio other than $1$, is not a perpendicular bisector. It is still a circle, called an Apollonius circle, and the past question above shows the standard method: write $z = x + iy$, square both sides to clear the moduli, and complete the square. The ratio $\sqrt{2}$ shifts the centre off the midpoint and sets a finite radius. Recognising that an unequal ratio of moduli gives a circle (while an equal ratio gives a line) saves you from expecting the wrong shape.