Question:
If arg (z – 1) = arg (z + 3i), then find x – 1 : y. where z = x + iy
Solution:
According to the question,
Let z = x + iy
Given that,
arg (z – 1) = arg (z + 3i)
⇒ arg (x + iy – 1) = arg (x + iy + 3i)
⇒ arg (x – 1 + iy) = arg (x + I (y) = π/4
$\Rightarrow \tan ^{-1} \frac{y}{x-1}=\tan ^{-1} \frac{y+3}{x}$
$\Rightarrow \frac{y}{x-1}=\frac{y+3}{x}$
⇒ xy = xy – y + 3x – 3
⇒ 3x – 3 = y
⇒ (x – 1)/y = 1/3
Hence, (x – 1): y = 1: 3