question_answer

If \[A\,(1+i),\]\[B\,(3+4i)\] and \[C(z)\] are the vertices of a \[\Delta \,ABC\] in which \[\angle BAC=\frac{\pi }{3}\] and \[AC=2AB.\] Then z is-

A) \[3+4i+i\sqrt{3}(2+3i)\]

B) \[(3+4i)+\frac{1}{\sqrt{3}}\,\,i(2+3i)\]

C) \[(2+3i)+i\sqrt{3}(3+4i)\]

D) \[(2+3i)+\frac{i}{\sqrt{3}}(3+4i)\]

Correct Answer: A

Solution :

\[\frac{z-(1-i)}{AC}=\frac{(2+3i)}{AB}{{e}^{{{i}^{\pi /3}}}}\]

\[\Rightarrow z-1-i=2\,\,(2+3i)\left( \frac{1}{2}+\frac{\sqrt{3}}{2}i \right)\]

\[\Rightarrow z-1-i=(2+3i)\,\,(1+\sqrt{3}i)\]

\[\Rightarrow z=3+4i+i\sqrt{3}(2+3i)\]