Question

$ABCD$ is a rhombus. Its diagonals $AC$ and $BD$ intersect at the point $M$ and satisfy $BD=2AC$. If the points $D$ and $M$ represent the complex numbers $1+i$ and $2-i$, respectively, then $C$ represents the complex numbers

• A. $3-\frac{i}{2}$
• B. $1-\frac{3}{2}i$
• C. $1+\frac{3}{2}i$
• D. $3+\frac{i}{2}$

Answer 1

Answer: (A), (B)

$1-\frac{3}{2}i$


Rotating $DM$ about $M$ by an angle ${90}^{\circ },$ we have
$\frac{z-(2-i)}{(1+i)-(2-i)}=\frac{|z-(2-i)|}{|(1+i)-(2-i)|}{e}^{\pm i\frac{\pi }{2}}$
$\Rightarrow \frac{z-(2-i)}{-1+2i}=\pm \frac{i}{2}$
$\Rightarrow 2z=(-i-2)+(4-2i)\text{or}(i+2)+(4-2i)$
$\Rightarrow z=1-\frac{3}{2}i\text{or}3-\frac{i}{2}$