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Show that the complex number z , satisfy...

Show that the complex number `z ,` satisfying are `(z-1)/(z+1)=pi/4` lies on a circle.

Text Solution

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Let z = x +`i`y
Given that, arg` ((z - 1)/(z + 1)) = pi//4`
`rArr arg(z -1) -arg (z + 1) = pi//4`
`rArr arg(x +iy - 1)- arg (x + iy + 1) = pi //4`
`rArr arg(x - 1 + iy)- arg ( x + 1 + iy) = pi/4`
`rArr tan^(-1) (y)/(x - 1) - tan^(-1) (y)/(x + 1) = pi//4`
`rArr tan^(-1)[((y)/(x -1)-(y)/(x +1))/ (1 +((y)/(x -1))((y)/(x+1)))]=pi//4`
`rArr [y[(x + 1- x +1)/(x^(2) - 1)]]/((x ^(2)-1 + Y^(2))/(x^(2)-1))= tan pi //4`
`rArr (2y)/(x^(2) + Y^(2) - 1) = 1`
`rArr x^(2) + y^(2) - 1 = 2 y`
`rArr x^(2) + Y^(2) - 1 = 0` which represents a circle .
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