24Find the locus of a as z travels on real axis from -∞ to +∞.
If a=(z-i)/(z+i) then show that when z lies above the real axis , a will lie within the unit circle which has the centre at origin.Find the locus of a as z travels on real axis from -∞ to +∞.
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3 Answers
62|(z-i)/(z+i)|
z=a+ib
b>0
so |z-i|2=a2+(b-1)2
so |z+i|2=a2+(b+1)2
clearly b+1>b-1 (b>0)
so |z-i|<|z+i|
so the fraction has modulus <1
hence it lies in a unit circle :)
62z is on the real axis.. so b=0
|(z-i)/(z+i)| = 1
so this will give a circle of radius 1