the roots of the equation z³+az²+bz+c=0 (where a,b,c are complex numbers )are the vertices of an equilateral triangle in the Argand plane.if ab=9(c+1) find the area of the triangle

Let the roots be p,q,r.

We know that if p,q,r are the vertices of an equilateral triangle then the following relation holds.

p² + q²+r² = pq+qr+rp. In terms of the coefficients we get a²=3b. We are further given that ab=9(c+1).

Substituting for b,c we can write the equation as z³+az²+a²z/3 + (a³/27 - 1) = 0 or (z+a/3)³ = 1

Hence, we see that by translating the origin to z=-a/3, p,q,r form an equilateral triangle inscribed in a circle of radius 1.

Such a triangle has an area 3√3/4