Let z 1 , z 2 be the roots of the equation z 2 + a z + 12 = 0 and z 1 , z 2 form an equilateral triangle with origin. Then, the value of a is

Step-by-step solution

z 2 + a z + 12 = 0 z 1 + z 2 = - a and z 1 z 2 = 12 If 0 ,   z 1 ,   z 2 are vertices of equilateral triangles z 2 = z 1 e i π 3 z 2 = z 1 1 2 + i 3 2 2 z 2 - z 1 = 3 i z 1 squaring both sides ⇒ 4 z 2 2 + z 1 2 - 4 z 1 z 2 = - 3 z 1 2 ⇒ z 1 + z 2 2 = 3 z 1 z 2 ⇒ a 2 = 3 × 12 ⇒ a = 6