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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. |
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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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