Let z be a complex number such that |z-2i|=3 and Re ( z+i z-2i )= 2 3 . If z₁ and z₂ are the two possible values of z satisfying these conditions, then the distance |z₁-z₂| is equal to

Step-by-step solution

From the given condition |z-2i|=3 , the complex number z lies on a circle of radius 3 centered at 2i . Let z-2i = 3( + i ) . Then z+i = (z-2i) + 3i = 3( + i ) + 3i = 3 + i(3 + 3) . Substitute this into the given rational expression: z+i z-2i = 3 + i(3 + 3) 3( + i ) z+i z-2i = ( + i( + 1))( - i ) z+i z-2i = ^2 + ( + 1) + i( ( + 1) - ) z+i z-2i = ( ^2 + ^2 + ) + i = (1 + ) + i The real part is 1 + . We are given that the real part is 2 3 . 1 + = 2 3 = - 1 3 Since ^2 = 1 - ^2 , we have = 1 - 1 9 = 2 2 3 . Now, substit