Question

The complex solutions of (z + i) 2011 = z 2011 lie on: A. A circle. B. An ellipse. C. A hyperbola. D. A straight line.

Answer 1

$let\ \ z = x+ j y\\\\LHS=[x+j(y+1)]^{1011}\\\\RHS=(x+iy)^{1011}\\\\Magnitude\ of\ LHS=[x^2+(y+1)^2]^{1011/2}\\\\Magnitude\ of\ RHS=(x^2+y^2)^{1011/2}\\\\equating\ them:\ y^2=y^2+2y+1\\\\y=-1/2$

Thus the locus of the points is a straight line parallel to x axis.