Question

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

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

Answer 1

Answer: (D)

$(3-\frac{1}{2}i)$ or $(1-\frac{3}{2}i)$

$\frac{z-(2-i)}{1+i-(2-i)}=\frac{AM}{MD}{e}^{\frac{2\pi i}{2}}$
$\Rightarrow \frac{z-(2-i)}{-i+2i}=\frac{1}{2}(\pm i)$
$\Rightarrow z=(2-i)+\frac{i}{2}(-1+2i)$
$z=\left(2i\right)\frac{-i}{2}(-1+2i)$
$\Rightarrow z=1-\frac{3}{2}i$ or $z=3-\frac{1}{2}i$