Question
Question: The roots z<sub>1</sub>, z<sub>2</sub>, z<sub>3</sub> of the equation x<sup>3</sup>+ 3px<sup>2</sup>...
The roots z1, z2, z3 of the equation x3+ 3px2 + 2qx + r = 0 (p, q, r are complex numbers) correspond to points A, B and C. Then triangle ABC is equilateral if –
A
p = q2
B
p2 = 3q
C
p2 = q
D
q2 = 3p
Answer
p2 = q
Explanation
Solution
Sol. For equilateral D
z12+z22+z32= z1z2+z2z3+z3z1
(z1+z2+z3)2=3(z1z2+z2z3+z3z1) (– 3p)2 = 3 × 3q
p2 = q