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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. |
Answer» <html><body><p></p>Solution :We note that the quotient `(<a href="https://interviewquestions.tuteehub.com/tag/z-750254" style="font-weight:bold;" target="_blank" title="Click to know more about Z">Z</a>-i)/(z-1)` is not <a href="https://interviewquestions.tuteehub.com/tag/defined-947013" style="font-weight:bold;" target="_blank" title="Click to know more about DEFINED">DEFINED</a> if `z=1` <br/> Sine `z=x I y`<br/> `(z-i)/(z-1)=(xiy-i)/(x+iy-1)` <br/> `=(x+i(y-1))/(x-1+iy)=([(x+i(y-1)][(x-1)-iy)])/([(x-1)-iy])` <br/> `=((x^(<a href="https://interviewquestions.tuteehub.com/tag/2-283658" style="font-weight:bold;" target="_blank" title="Click to know more about 2">2</a>)+y^(2)-x-y)/((x-1)^(2)+y^(2)))+i((1-x+y)/((x-1)^(2)+y^(2)))` <br/>`(z-i)/(z-1)` is purelyimaginry if Re part =0 <br/> `<a href="https://interviewquestions.tuteehub.com/tag/harr-2692945" style="font-weight:bold;" target="_blank" title="Click to know more about HARR">HARR</a> z ne 1 and (x^(2)+y^(2)-x-y)/((x-1)^(2)+y^(2))=0` <br/> `harr x^(2)+y^(2)-x-y=0` <br/>and `(x,y) ne (1,0)` <br/>`:.` The locus of P is the <a href="https://interviewquestions.tuteehub.com/tag/circle-916533" style="font-weight:bold;" target="_blank" title="Click to know more about CIRCLE">CIRCLE</a><br/> `x^(2)+y^(2)-x-y=0` <br/> excluding the pint (1,0)</body></html> | |