The locus of a point on the argand plane represented by the complex number z , when z satisfies the condition | z-1+i z+1-i |= | Re ( z-1+i z+1-i ) | is

Step-by-step solution

Given condition is | z-1+i z+1-i |= | Re ( z-1+i z+1-i ) |, z -1+i Let z=x+i y , then aligned & z-1+i z+1-i = (x-1)+i(y+1) (x+1)+i(y-1) (x+1)-i(y-1) (x+1)-i(y-1) & = [ (x^2-1 )+ (y^2-1 ) ]+i[(x+1)(y+1)-(x-1) (x+1)^2+(y-1)^2 aligned So, | z-1+i z+1-i |= [ (x^2-1 )+ (y^2-1 ) ]^2+(2(x+y))^2 ((x+1)^2+(y-1)^2 )^2 and | Re ( z-1+i z+1-i ) |= | (x^2-1 )+ (y^2-1 ) | (x+1)^2+(y-1)^2 Now, according to given condition aligned & ( (x^2-1 )+ (y^2-1 ) )^2+(2(x+y))^2 (x+1)^2+(y-1)^2 & = | (x^2-1 )+ (y^2-1 ) | (x+1)^2+(y-1)^2 & (