First slide

Geometry of complex numbers

Question

$Q (z_{q})$ is the foot of perpendicular drawn from $P (z_{p})$ to the line $a \bar{z} + z \bar{a} + b = 0,$ $b \in R$ and $' a'$ is complex number, then:

Moderate

A

$z_{q} \bar{a} + a {\bar{z}}_{p} - z_{p} \bar{a} + b = 0$
B

$z_{q} \bar{a} + 2 a {\bar{z}}_{p} - z_{p} \bar{a} + b = 0$
C

$2 z_{q} \bar{a} + a {\bar{z}}_{p} - z_{p} \bar{a} + b = 0$
D

$z_{q} \bar{a} + a {\bar{z}}_{p} - 2 z_{p} \bar{a} + b = 0$

Solution

Since a is perpendicular vector to the line.
$z_{q} - z_{p} = λ a, λ \in R$
$\bar{a} (z_{p} + λ a) + a ({\bar{z}}_{p} + λ \bar{a}) + b = 0$
$\Rightarrow$      $a {\bar{z}}_{p} + \bar{a} z_{p} + b + 2 λ a \bar{a} = 0$
Also $2 \bar{a} z_{q} - 2 \bar{a} z_{p} = 2 λ a \bar{a}$
$∴$        $a {\bar{z}}_{p} + \bar{a} z_{p} + b + 2 \bar{a} z_{q} - 2 \bar{a} z_{p} = 0$
$\Rightarrow$    $2 \bar{a} z_{q} + a {\bar{z}}_{p} - \bar{a} z_{p} + b = 0$

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