Question

Let $O,A,B$ be three collinear points such that $OA.OB=1$. If $O$ and $B$ represent the complex numbers $o$ and $z$ then $A$ represents

• A. $\frac{1}{\overline{z}}$
• B. $\frac{1}{z}$
• C. $\overline{z}$
• D. $z^2$

Answer 1

The correct option is A $\frac{1}{\overline{z}}$

Let A represent $z_1$

SInce $OA.OB=1$ so, $|z_1-0|.|z-0|=1\Rightarrow \left|z_1\right|=\frac{1}{\left|z\right|}$

Also, $arg\left(\frac{z_1-0}{z-0}\right)=0\Rightarrow arg\left(\frac{z_1}{z}\right)=0\Rightarrow arg\left(z_1\right)=arg\left(z\right)$

It $\theta$ is the argument of $z$ then $z=\left|z\right|e^{i\theta }$

thus, $z_1=\frac{1}{\left|z\right|}{e}^{i\theta }=\frac{1}{{\left|z\right|}^2}\left|z\right|e^{i\theta }=\frac{1}{\overline{z}}$

Thus A is $\frac{1}{\overline{z}}$