Question

All the points in the set `S={(alpha+i)/(alpha-i):alpha in R } (i= sqrt(-1))lie` on a
A. cicle whose radius is `sqrt(2)`
B. straight line whose slope is -1
C. circle whose radius is 1.
D. straight line whose slope is 1.

Answer 1

Correct Answer - C
Let x+I y = `(alpha +i)/(alpha- i)`
`rArr x+iy=((alpha+i)^2)/(alpha ^2+1) and y=(2 alpha )/(alpha^2n+1)`
Now `x^2+y^2=((alpha^2-1)/(alpha ^2+1))^2+((2alpha )/(alpha)^2+I)^2`
`=(alpha^4+1-2alpha^2+4 alpha^2)/(alpha^2 +1)^2=(alpha ^2+1)^2/(alpha^2+1)^2=1`
`rArr x^2+y^2=1`
Which is an equations of circle with centre (0,0) and radius 1 unit . So,`S={(alpha+i)/(alpha-1),alpha ne R}` lies on a circle with radius 1.