Complex Numbers And Quadratic Equations question 356
Question: The region of the complex plane for which $ | \frac{z-a}{z+\overline{a}} |=1 $ $ [R(a)\ne 0] $ is
Options:
A) $ x- $ axis
B) $ y- $ axis
C) The straight line $ x=a $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
We have $ | \frac{z-a}{z+\bar{a}} |=1 $
Þ $ |z-a|=|z+\overline{a}| $
Þ $ |z-a{{|}^{2}}=|z+\overline{a}{{|}^{2}} $
Þ $ (z-a)(\overline{z-a})=(z+\overline{a})(\overline{z+\overline{a}}) $
Þ $ (z-a)(\overline{z}-\overline{a})=(z+\overline{a})(\overline{z}+a) $
Þ $ z\overline{z}-z\overline{a}-a\overline{z}+a\overline{a}=z\overline{z}+za+\overline{a}\overline{z}+\overline{a}a $
Þ $ za+z\overline{a}+\overline{a}\overline{z}+a\overline{z}=0\Rightarrow (a+\overline{a})(z+\overline{z})=0 $
Þ $ z+\overline{z}=0(\because a+\overline{a}=2Re(a)\ne 0) $
Þ $ 2Re(z)=0 $
Þ $ 2x=0 $
Þ $ x=0 $ Which is the equation of y-axis.
BETA
