If z is a complex number, then curves |z|=1 , |z-2|=1 and |z-1|=0 have a common point at

Step-by-step solution

aligned & |z|=1,|z-2|=1,|z-1|=0 & |z|=1 x^2+y^2=1...(i) & |z-2|=1 (x-2)^2+(y)^2=1 ...(ii) & |z-1|=0 (x-1)^2+y^2=0...(iii) aligned Solving Eqs. (i) and (ii), we get aligned (x-2)^2-x^2 & =0 (x-2)^2 & =x^2 x-2= x aligned Taking positive aligned x-2 & =x -2 & =0 (Not true) aligned Taking negative aligned x-2 & =-x 2 x & =2 x & =1 y^2=1-x^2=1-1=0 y & =0 aligned Point of intersection of Eqs. (i) and (ii) is (1,0) . Now, put it in Eq. (iii), we get aligned (1-1)^2+0^2 & =0 0 & =0 aligned It also lies on circle of Eq. (ii