Question

POQ is a straight line through origin of O, P and Q represent the complex numbers $a+ib$ and $c+id$ respectively and OP = OQ, then,
1. $\left| a+ib \right|=\left| c+id \right|$
2. $a+c=b+d$
3. $\arg \left( a+ib \right)=\arg \left( c+id \right)$
4. None of these

Answer 1

For solving this question you should know the concept of straight line. Here the origin and both the points are given to us, so we will use the property of midpoint and it will be OP = OQ and then by applying the coordinates of origin, we will get the answer for the relation of both the points.

Complete step-by-step solution:
According to the question a straight line POQ is given to us and the origin and points P and Q represent the complex numbers $a+ib$ and $c+id$ respectively. And it is also given to us that OP = OQ. As we know that POQ is the straight line here. And the origin is given by O. So, the origin,
$\begin{aligned} & O=0+i0 \\\ & P=a+ib \\\ & Q=c+id \\\ \end{aligned}$
Since OP = OQ is given to us, it means that origin O is the midpoint for the straight line POQ. So, it can be written as,
$a+ib=c+id$
The coordinates of O (0,0), so,
$\dfrac{a+c}{2}=0,\dfrac{b+d}{2}=0$
So, it means that,
$a+c=b+d$
Hence the correct answer is option 2.

Note: While solving any question in a straight line you have to always keep in mind that the given points are always satisfying the equation if there is any equation. And this condition is applied for all the curves too. It means that if we put the points in the equation then we find that the equation is solved completely and accurately.