Complex Numbers And Quadratic Equations question 332
Question: If complex numbers $ z_1,z_2\text{and }z_3 $ represent the vertices A, B and C respectively of an isosceles triangle ABC of which $ \angle C $ is right angle, then correct statement is [RPET 1999]
Options:
A) $ z_1^{2}+z_2^{2}+z_3^{2}=z_1z_2z_3 $
B) $ {{(z_3-z_1)}^{2}}=z_3-z_2 $
C) $ {{(z_1-z_2)}^{2}}=(z_1-z_3)(z_3-z_2) $
D) $ {{(z_1-z_2)}^{2}}=2(z_1-z_3)(z_3-z_2) $
Show Answer
Answer:
Correct Answer: D
Solution:
$ BC=AC $ and $ \angle C=\pi /2 $ By rotation about $ C $ in anti-clockwise sense $ CB=CA{e^{i\pi /2}} $
Þ $ (z_2-z_3)=(z_1-z_3){e^{i\pi /2}}=i(z_1-z_3) $
Þ $ {{(z_2-z_3)}^{2}}=-{{(z_1-z_3)}^{2}} $
Þ $ z_2^{2}+z_3^{2}-2z_2z_3=-z_1^{2}-z_3^{2}+2z_1z_3 $
Þ $ z_1^{2}+z_2^{2}-2z_1z_2=2z_1z_3+2z_2z_3-2z_3^{2}-2z_1z_2 $
Þ $ {{(z_1-z_2)}^{2}}=2[(z_1z_3-z_3^{2})-(z_1z_2-z_2z_3)] $
Þ $ {{(z_1-z_2)}^{2}}=2(z_1-z_3)(z_3-z_2) $ .
BETA
