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the roots of the equation z 3 +az 2 +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

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

Grade:11

1 Answers

mycroft holmes
272 Points
11 years ago

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

 


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