question_answer

\[ABCD\] is a rhombus. Its diagonals \[AC\] and \[BD\] intersect at the point \[M\] and satisfy \[BD=2AC\].  If the points \[D\] and \[M\] represents the complex numbers \[1+i\] and \[2-i\] respectively, then \[A\] represents the complex number

A) \[3-\frac{1}{2}i\]or \[1-\frac{3}{2}i\]

B) \[\frac{3}{2}-i\]or \[\frac{1}{2}-3i\]

C) \[\frac{1}{2}-i\]or \[1-\frac{1}{2}i\]

D) None of these

Correct Answer: A

Solution :

\[BD=2AC\Rightarrow 2DM=2(2AM)\] or \[DM=2AM\]or \[D{{M}^{2}}=4A{{M}^{2}}\] or \[5=4[{{(x-2)}^{2}}+{{(y+1)}^{2}}]\]            .....(i) Again slope of \[DM=-2\] and slope of \[AM\]is \[\frac{y+1}{x-2}\] AM is perpendicular to DM \[\therefore \,\,\,-2\left( \frac{y+1}{x-2} \right)=-1\Rightarrow x-2=2(y+1)\]        .....(ii) Hence from (i) and (ii), we get \[\therefore \,\,y=-\frac{1}{2},-\frac{3}{2}\]and \[x=3,1\]