Complex Numbers
Algebra of Complex Numbers
Grade Class 11
Question:
<p>If \(z\) satisfies \(\arg(z-2)=\pi/6\) and \(\arg(z-2i)=2\pi/3\), then \(z\) is:</p>
\(2+2i\)
\(2+i\sqrt{3}\)
\(\sqrt{3}+2i\)
\(1+i\sqrt{3}\)
Step-by-Step Solution
Key Concept: Each arg condition gives a half-line. Find their intersection by solving the two line equations simultaneously.
<p>From $\arg(z-2)=\pi/6$: $y/(x-2)=\tan(\pi/6)=1/\sqrt{3}\Rightarrow y=(x-2)/\sqrt{3}$. From $\arg(z-2i)=2\pi/3$: $(y-2)/(x)=\tan(2\pi/3)=-\sqrt{3}\Rightarrow y-2=-\sqrt{3}x$. Solve simultaneously to get $z$.</p>
Correct Answer: BD