Question

Let $O$ be the origin and $A$ be the point $z _1=1+2 i$. If $B$ is the point $z _2, \operatorname{Re}\left( z _2\right)<0$, such that $OAB$ is a right angled isosceles triangle with $OB$ as hypotenuse, then which of the following is NOT true ?

• A. $\arg z _2=\pi-\tan ^{-1} 3$
• B. $\arg \left( z _1-2 z _2\right)=-\tan ^{-1} \frac{4}{3}$
• C. $\left|z_2\right|=\sqrt{10}$
• D. $\left|2 z_1-z_2\right|=5$

Answer 1

Answer: (D)

$AB = AO \cdot z ^{-i \pi / 2}=-2+i$
So $OB =(-2+i)+(1+2 i)$
$ z _2=-1+3 i $
$ \therefore\left|2 z _1- z _2\right|=\sqrt{10}$