Complex Numbers And Quadratic Equations question 326
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 $ represents the complex numbers $ 1+i $ and $ 2-i $ respectively, then $ A $ represents the complex number
Options:
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
Show Answer
Answer:
Correct Answer: A
Solution:
$ BD=2AC\Rightarrow 2DM=2(2AM) $ or $ DM=2AM $ or $ DM^{2}=4AM^{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( \frac{y+1}{x-2} )=-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 $
BETA
