JEE Main 2022 (Online) 26th July Morning ShiftQuestion 1
Let $O$ be the origin and $A$ be the point $z_1=1+2i$. If $B$ is the point $z_2$, $\mathrm{Re}(z_2)<0$, such that $OAB$ is a right angled isosceles triangle with $OB$ as hypotenuse, then which of the following is NOT true ?
Complex Numbers
Solution
Right angle at $A$, so $z_2=z_1+(\pm i)z_1=z_1(1\pm i)$: $-1+3i$ or $3+i$. $\mathrm{Re}(z_2)<0\Rightarrow z_2=-1+3i$. (A) $\arg z_2=\pi-\tan^{-1}3$ true. (C) $|z_2|=\sqrt{10}$ true. $z_1-2z_2=3-4i$, $\arg=-\tan^{-1}\frac43$ true (B). $2z_1-z_2=3+i$, modulus $\sqrt{10}\ne5$, so (D) is NOT true. Answer: (D).