(A (z₁ ) ) and (B (z₂ ) ) are two points in the argand plane. Then, the locus of the complex number (z ) satisfying ( ( z-z₁ z-z₂ )=0 ) or ( ), is

Step-by-step solution

Let the point (z(x, y), z₁ (x₁, y₁ ) ) and (z₂ (x₂, y₂ ) ). ( aligned & z-z₁=x+i y-x₁-i y₁ & =x-x₁+i (y-y₁ ) & and z-z₂=x+i y-x₂-i y₂ & =x-x₂+i (y-y₂ ) aligned ) Now, ( z-z₁ z-z₂ ) ( aligned & = [ (x-x₁ )+i (y-y₁ ) ] [ (x-x₂ )+i (y-y₂ ) ] [ (x-x₂ )-i (y-y₂ ) ] [ (x-x₂ )-i (y-y₂ ) ] & (x-x₁ ) (x-x₂ )+ (y-y₁ ) (y-y₂ )+ & = i [ (x-x₂ ) (y-y₁ )- (x-x₁ ) (y-y₂ ) ] (x-x₂ )^2+ (y-y₂ )^2 & = Arg ( z-z₁ z-z₂ )=0 or & [ [ (x-x₂ ) (y-y₁ ) ]- [ (x-x₁ ) (y-y₂ ) ] [ (x-x₁ ) (x-x₂ )+ (y-y₁ ) (y-y₂ ) ] ]=0 & (x-x₂ ) (y-y₁ )= (x-x₁