The complex numbers $z=x+iy$ which satisfy the equation $\left|\dfrac{z-5i}{z+5i}\right|=1$ lie on
Step-by-Step Solution
Key Concept: $|z-a|=|z-b|$ is always the perpendicular bisector of $ab$. Here $a=5i$ and $b=-5i$ are symmetric about the real axis, so the bisector is the $X$-axis.
**Step 1: Simplify the equation**
$\left|\dfrac{z-5i}{z+5i}\right|=1 \Rightarrow |z-5i|=|z+5i|$.
**Step 2: Interpret geometrically**
$z$ is equidistant from $5i=(0,5)$ and $-5i=(0,-5)$. The locus is the perpendicular bisector of the segment joining them, which is the $X$-axis ($y=0$).
**Step 3: Verify algebraically**
Let $z=x+iy$: $x^2+(y-5)^2=x^2+(y+5)^2 \Rightarrow -20y=0 \Rightarrow y=0$. ✓
Correct Answer: 1