question_answer

If P is the point in the Agrand diagram corresponding to the complex number \[\sqrt{3}+i\] and if OPQ is an isosceles right angled triangle, right angled at ?O?, then Q represents the complex number

A)  \[-1+i\sqrt{3}\] or \[1-i\sqrt{3}\]

B)  \[1\pm i\sqrt{3}\]

C)  \[\sqrt{3}-i\] or \[1-i\sqrt{3}\]

D)  \[-1\pm i\sqrt{3}\]

Correct Answer: A

Solution :

Let \[z=\sqrt{3}+i\] \[\therefore \]  \[\arg (z)={{\tan }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\]                 \[={{30}^{o}}\] For making a right angled triangle OPQ, point Q either in II nd quadrant or IVth quadrant. If the point Q is in IInd quadrant, then we take \[\theta ={{120}^{o}}\] \[\therefore \]  \[\tan {{120}^{o}}=-\cot {{30}^{o}}\]                                 \[=\frac{\sqrt{3}}{-1}\] \[\therefore \]  Point Q is \[(-1,\sqrt{3)}\] and if the point Q is in IVth quadrant, then we take                 \[\theta =-{{60}^{o}}\] \[\therefore \]  \[\tan (-{{60}^{o}})=-\tan {{60}^{o}}\]                                 \[=-\frac{1}{\sqrt{3}}\] \[\therefore \]  Point  Q is \[(1,-\sqrt{3})\] Hence, option [a] is correct.