Let O be the origin and A be the point z_1=1+2i.

Question

Let $O$ be the origin and $A$ be the point $z_1=1+2i$. If $B$ is the point $z_2$, $\operatorname{Re}(z_2)<0$, such that $OAB$ is a right-angled isosceles triangle with $OB$ as the hypotenuse, then which of the following is NOT true?

Accepted Answer

Rotating $z_1$ by $45^{\circ}$ (and scaling): $z_2=(1+2i)(1+i)=1+i+2i-2=-1+3i$.

$\arg(z_2)$: point $(-1,3)$ is in the second quadrant, so $\arg z_2=\pi-	an^{-1}3$ --- (A) true.

$|z_2|=\sqrt{1+9}=\sqrt{10}$ --- (C) true.

$z_1-2z_2=(1+2i)-2(-1+3i)=3-4i$, so $\arg(z_1-2z_2)=-	an^{-1}\dfrac43$ --- (B) true.

$2z_1-z_2=(2+4i)-(-1+3i)=3+i$, so $|2z_1-z_2|=\sqrt{9+1}=\sqrt{10}
eq5$ --- (D) is FALSE.

Solutions · 1