The roots `z_1, z_2, z_3` of the equation `x^3 + 3ax^2 + 3bx + c = 0` in which a, b, c are complex numbers correspond to points A, B, C. Show triangle will be an equilateral triangle if `a^2=b`.

If the equation ax^2 + bx + c = 0, 0 < a < b < c, has non real complex roots z_1 and z_2 then

If the equation x^(2 )+ 2x + 3 = 0 and ax^(2) +bx+c=0, a, b, c in R , have a common root, then a : b:c is

If the equations ax^2 + bx + c = 0 and x^3 + x - 2 = 0 have two common roots then show that 2a = 2b = c .

If the complex numbers z_1,z_2,z_3 represents the vertices of a triangle ABC, where z_1,z_2,z_3 are the roots of equation z^3+3alphaz^2+3alphaz^2+3betaz+gamma=0, alpha,beta,gamma beng complex numbers and alpha^2=beta then /_\ABC is (A) equilateral (B) right angled (C) isosceles but not equilateral (D) scalene

For a complex number Z, if all the roots of the equation Z^3 + aZ^2 + bZ + c = 0 are unit modulus, then

The roots of the equation t^3+3a t^2+3b t+c=0a r ez_1, z_2, z_3 which represent the vertices of an equilateral triangle. Then a^2=3b b. b^2=a c. a^2=b d. b^2=3a

If z_1,z_2,z_3 are the vertices of the A B C on the complex plane and are also the roots of the equation z^3-3a z^2+3betaz+gamma=0 then the condition for the A B C to be equilateral triangle is: alpha^2=beta (b) alpha=beta^2 alpha^2=3beta (d) alpha=3beta^2

a and b are real numbers between 0 and 1 such that the points Z_1 =a+ i , Z_2=1+ bi , Z_3= 0 form an equilateral triangle, then a and b are equal to

If a, b, c are positive and a = 2b + 3c, then roots of the equation ax^(2) + bx + c = 0 are real for

The triangle formed by complex numbers z ,i z ,i^2z is (a)Equilateral (b) Isosceles (c)Right angle (d) Scalene