alleryStart learning →The locus of the point $z$ in the Argand plane satisfying $|z - 2| = |z + 2|$ is:
- A parabola
- A circle of radius $2$ centred at the origin
- The real axis ($\mathrm{Im}(z) = 0$)
- The imaginary axis ($\mathrm{Re}(z) = 0$)
Show answer and solution
Correct answer: D. The imaginary axis ($\mathrm{Re}(z) = 0$)
**Geometric reading.** $|z - 2|$ is the distance from the point $z$ to $(2, 0)$. $|z + 2|$ is the distance to $(-2, 0)$. Equating them means $z$ is equidistant from the two points.
**Locus.** Set of points equidistant from $(2, 0)$ and $(-2, 0)$ is the **perpendicular bisector** of the segment joining them — the $y$-axis, i.e. $\mathrm{Re}(z) = 0$.
**Algebraic confirmation.** Let $z = x + iy$. Squaring: $(x-2)^2 + y^2 = (x+2)^2 + y^2 \Rightarrow -4x = 4x \Rightarrow x = 0$.
**Why option C is tempting.** A symmetric-looking radius-2 hint, but $|z - 2|$ is *not* the same as $|z| - 2$
it's a distance to a *point*, not a centred circle equation.