Complex Numbers
Conjugate and Modulus – Geometric Locus
Complex Numbers_PYQ
Grade 11
Question:
The equation $|z - i| = |z - 1|$, $i = \sqrt{-1}$, represents
a circle of radius $\dfrac{1}{2}$
the line passing through the origin with slope $1$
a circle of radius $1$
the line passing through the origin with slope $-1$
Step-by-Step Solution
Key Concept: $|z-a|=|z-b|$ is the perpendicular bisector of the segment joining the points $a$ and $b$ in the complex plane. Here $a=i=(0,1)$ and $b=1=(1,0)$; their perpendicular bisector has slope $1$.
**Step 1: Write z in component form**
Let $z = x + iy$. Then $|z-i|^2 = x^2+(y-1)^2$ and $|z-1|^2 = (x-1)^2+y^2$.
**Step 2: Set moduli equal and simplify**
$x^2+(y-1)^2=(x-1)^2+y^2 \Rightarrow x^2+y^2-2y+1=x^2-2x+1+y^2 \Rightarrow -2y=-2x \Rightarrow y=x$.
**Step 3: Identify the locus**
$y = x$ is a straight line through the origin with slope $1$.
Correct Answer: 2