Question

$POQ$ is a straight line through of origin $O,P$ and $Q$ represent the complex numbers $a+ib$ and $c+id$ respectively and $OP=OQ$ ,then

• A. $\left|a+ib\right|=\left|c+id\right|$
• B. $a+c=b+d$
• C. $arg(a+ib)=arg(c+id)$
• D. None of these

Answer 1

The correct option is B

$a+c=b+d$

Explanation for the correct option:

Using the midpoint formula on the given coordinates:

It is given that origin is the mid point of the line $POQ$ as $OP=OQ$.

So, coordinates of origin will be $\frac{(a+c)}{2}+\frac{(b+d)}{2}i$, but coordinates of the origin are given as $0+0i$.

On comparing we get, $\frac{a+c}{2}=0...\left(1\right)$ and $\frac{b+d}{2}=0...\left(2\right)$

Equating $\left(1\right)\mathrm{and}\left(2\right)$

$$\begin{gathered} \Rightarrow \frac{a+c}{2}=\frac{b+d}{2} \\ \Rightarrow a+c=b+d \end{gathered}$$

Hence, the correct answer is option (B)