Q42 / 125 · BITSAT 2017 · Mathematics

If $z = a + i b$ satisfies $\arg (z - 1) = \arg (z + 3 i)$ , then $(a - 1) : b =$

Solution

We have,

$\arg (z - 1) = \arg (z + 3 i)$

$\Rightarrow \arg ((a - 1) + i b) = \arg (a + (b + 3) i)$

$\Rightarrow \tan^{- 1} (\frac{b}{a - 1}) = \tan^{- 1} (\frac{b + 3}{a})$

$\Rightarrow \frac{b}{a - 1} = \frac{b + 3}{a}$

$\Rightarrow a b = (a - 1) (b + 3)$

$\Rightarrow 3 (a - 1) = b$

$\Rightarrow (a - 1) : b = 1 : 3$

Hence, the correct answer is option: 'B'.