1.

If the point P denotes the complexnumber z=x+ iy in the Argand plane and if (z-i)/(z-1) is a purelly imaginary number, find the locus of P.

Answer»

Solution :We note that the quotient `(Z-i)/(z-1)` is not DEFINED if `z=1`
Sine `z=x I y`
`(z-i)/(z-1)=(xiy-i)/(x+iy-1)`
`=(x+i(y-1))/(x-1+iy)=([(x+i(y-1)][(x-1)-iy)])/([(x-1)-iy])`
`=((x^(2)+y^(2)-x-y)/((x-1)^(2)+y^(2)))+i((1-x+y)/((x-1)^(2)+y^(2)))`
`(z-i)/(z-1)` is purelyimaginry if Re part =0
`HARR z ne 1 and (x^(2)+y^(2)-x-y)/((x-1)^(2)+y^(2))=0`
`harr x^(2)+y^(2)-x-y=0`
and `(x,y) ne (1,0)`
`:.` The locus of P is the CIRCLE
`x^(2)+y^(2)-x-y=0`
excluding the pint (1,0)


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