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Q.

Equation of straight line on the complex plane passing through a point ‘P’ denoting the complex number $' α'$ and perpendicular to the vector $\vec{O P}$ where ‘O’ is origin can be _____

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a

$Re (\bar{α} z) = 0$

b

$\bar{α} z + α \bar{z} - 2 {| α |}^{2} = 0$

c

$Re (\frac{z - α}{α}) = 0$

d

$Im (\frac{z - α}{α}) = 0$

answer is B, D.

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Detailed Solution

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$\begin{array}{l} \arg (\frac{z - α}{α}) = \pm π / 2 \\ ∴ Re (\frac{z - α}{α}) = 0 \\ \Rightarrow \frac{1}{2} (\frac{z - α}{α} + \frac{\bar{z} - \bar{α}}{α}) = 0 \end{array} \begin{array}{l} \Rightarrow \frac{1}{2} (z \bar{α} - α \bar{α} + α \bar{z} - α \bar{α}) = 0 \\ \bar{α} z + α \bar{z} - 2 | α |^{2} = 0 \end{array}$

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Equation of straight line on the complex plane passing through a point ‘P’ denoting the complex number 'α' and perpendicular to the vector OP→ where ‘O’ is origin can be _____