Suppose that the complex number z lies on the curve such that z-4 z-2 i is purely imaginary. If the complex number z₁ represents the mid-point of chord OA of this curve, O being the origin, then z₁ necessarily satisfies

Step-by-step solution

z-4 z-2 i + z -4 z +2 i =0 which is a circle x^2+y^2-4 x-2 y=0 xx ₁+y y ₁-2 (x+x₁ )- (y+y₁ )=x₁^2+y₁^2-4 x₁-2 y₁ It passes through (0,0) , so, the locus of (x₁, y₁ ) is x^2+y^2-2 x-y=0 . So, z₁=x₁+i y₁ , lies on this circle for which the points (2,0) and (0,1) are extremities of a diameter. Also (0,0) and (2,1) represent extremities of another diameter