Let = x+iy be a non-zero complex number. The non-zero complex numbers z satisfying the equation z^2 + z = 0 form an equilateral triangle in the Argand plane. If the area of this triangle is 27 3 , then the value of x^2 + y^2 is equal to _______.

Step-by-step solution

Given the equation z^2 + z = 0 . For non-zero roots, multiply the equation by z to obtain z^3 + z z = 0 z^3 + |z|^2 = 0 . Taking the modulus on both sides gives |z|^3 = | | |z|^2 . Since z 0 , we get |z| = | | . Substituting |z|^2 = | |^2 back into the equation, we have z^3 = - | |^2 . The non-zero roots are the three cube roots of the complex number - | |^2 . These roots lie on a circle centered at the origin and are separated by an angle of 2 3 , thus forming an equilateral triangle. The radius of the circumcircl