Complex Numbers And Quadratic Equations question 379
Question: PQ and PR are two infinite rays. QAR is an arc. Point lying in the shaded region excluding the boundary satisfies [IIT Screening 2005]
Options:
A) $ |z-1|>2;|\arg (z-1)|<\frac{\pi }{4} $
B) $ |z-1|>2;|\arg (z-1)|<\frac{\pi }{2} $
C) $ |z+1|>2;|\arg (z+1)|<\frac{\pi }{4} $
D) $ |z+1|>2;|\arg (z+1)|<\frac{\pi }{2} $
Show Answer
Answer:
Correct Answer: C
Solution:
Equation of ray PQ $ \arg (z+1)=\frac{\pi }{4} $ Equation of ray PR $ \arg (z+1)=-\frac{\pi }{4} $ Shaded region is $ \frac{-\pi }{4}<\arg (z+1)<\frac{\pi }{4} $ $ |\arg (z+1)|<\frac{\pi }{4} $ ; $ |PQ|=\sqrt{{{(\sqrt{2})}^{2}}+{{(\sqrt{2})}^{2}}}=2 $ |PA| =2; |PR| = 2 so, arc QAR is of a circle of radius 2 unit with centre at $ P(-1,0) $ . All the points in the shaded region are exterior to this circle $ |z+1|=2 $ .
BETA
