Find the cartesian equation ofthe point p(x,y) representing the complex number z, given that #abs(z-1)=sqrt2abs(z-i# Show that the locus is a circle and state its radius and the coordinates of its center?

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Answer 1 · 2018-04-09T13:03:22

Given that the point P (x,y) represents the complex number z. and #abs(z-1)=sqrt2abs(z-i)# represents a circle.

So #z=x+iy#

Now

#abs(z-1)=sqrt2abs(z-i)#

#=>abs(x+iy-1)=sqrt2abs(x+i(y-1))#

#=>sqrt((x-1)^2+y^2)=sqrt2sqrt(x^2+(y-1)^2)#

#=>(x-1)^2+y^2=2(x^2+(y-1)^2)#

#=>x^2-2x+1+y^2=2x^2+2y^2-2y+2#

#=>x^2+y^2+2x-2y+1=0#

#=>(x+1)^2+(y-1)^2=1^2#

Obviously it is the locus of a circle and its radius #=1# and the coordinates of its center is #(-1,1)#