Complex Numbers And Quadratic Equations question 374
Question: If $ \frac{-31}{17} $ and $ \omega =\frac{1-iz}{z-i} $ than $ |\omega |=1 $ shows that in complex plane [RPET 1985, 97; IIT 1983; DCE 2000, 01; UPSEAT 2003; MP PET 2004]
Options:
A) z will be at imaginary axis
B) z will be at real axis
C) z will be at unity circle
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
$ w=\frac{1-iz}{z-i} $ , then $ |w|\ =1 $
Þ $ | \ \frac{1-iz}{z-i}\ |\ =1 $
Þ $ |1-iz|\ =\ |z-i| $
Þ $ |1-i(x+iy)|\ =\ |x+iy-i| $
Þ $ |(1+y)-ix|\ =\ |x+i(y-1)| $
Þ $ \sqrt{x^{2}+1+y^{2}+2y}=\sqrt{x^{2}+y^{2}+1-2y} $
Þ $ y=0 $ Hence $ z=x+iy=x $ . So z lies on real axis.
BETA
