Complex Numbers
Argument & Locus
MMTS_Full_Test_19
Grade 12
Question:
If $\arg\left(\dfrac{z-2}{z-2i}\right)=\dfrac{\pi}{4}$, then which of the following is correct?
$|z|_{\min}=0$
$|z|_{\min}=2(\sqrt{2}-1)$
$|z|_{\max}=2(\sqrt{2}+1)$
$|z|_{\max}=4$
Step-by-Step Solution
Key Concept: The locus is an arc of circle through $2$ and $2i$
Arc of circle through $(2,0)$ and $(0,2)$. Centre at $(1,1)$, radius$=\sqrt{2}$. Distance from origin to centre $=\sqrt{2}$. Min $|z|=\sqrt{2}-\sqrt{2}=0$... Wait: $|z|_{\min}=\sqrt{2}-\sqrt{2}=0$ on upper arc... $|z|_{\max}=2(\sqrt{2}+1)$ on one arc.
Correct Answer: 3