question_answer

The vertices \[B\] and \[D\] of a parallelogram are \[1-2i\]and \[4+2i\],  If the diagonals are at right angles and \[AC=2BD\], the complex number representing \[A\] is

A) \[\frac{5}{2}\]

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

C) \[3i-4\]

D) \[3i+4\]

Correct Answer: B

Solution :

We have \[|\overrightarrow{BD}|=|(4+2i)-(1-2i)|=\sqrt{9+16}=5\] Let the affix of A be \[z=x+iy\] The affix of the mid point of BD is \[\left( \frac{5}{2},0 \right)\]. Since the diagonals of a parallelogram bisect each other, therefore, the affix of the point of intersection of the diagonals is \[\left( \frac{5}{2},0 \right)\]. We have \[|\overrightarrow{AE}|=5\]  \[\left( \because BD=\frac{1}{2}AC=AE \right)\] Which is satisfied by option .