The sum of the square of the modulus of the elements in the set z= a + ib : a , b Z , z C ,|z-1| 1,|z-5| |z-5 i | is ________

Step-by-step solution

aligned & | z -1| 1 & |( x -1)+ iy | 1 & ( x -1)^2+ y ^2 1 & ( x -1)^2+ y ^2 1 (1) aligned aligned & Also |z-5| |z-5 i| & (x-5)^2+y^2 x^2+(y-5)^2 & -10 x -10 y & x y.....(2) aligned aligned & Solving (1) and (2) & array l ( x -1)^2+ x ^2=1 2 x ^2-2 x =0 x ( x -1)=0 x =0 or x =1 y =0 or y =1 array aligned Given x , y I Points (0,0),(1,0),(2,0),(1,1),(1,-1) to find aligned & |z₁ |^2+ |z₂ |^2+ |z₃ |^2+ |z₄ |^2+ |z₅ |^2 & =0+1+4+1+1+1+1=9 aligned