Given quadratic equation, Z2+aZ+b=0 and two roots are Z1 and Z2. ∴Z1+Z2=−a and Z1Z2=b Question says, There are three complex numbers:
1. Origin (0)
2. Z1
3. Z2
and they form an equilateral triangle. So They are the vertices of the triangle. [ Important Point : If Z1,Z2 and Z3 are the vertices of an equilateral triangle then - Z12+Z22+Z32=Z1Z2+Z2Z3+Z3Z1] In this question, Z1=0,Z2=Z1 and Z3=Z2 By putting those values in the equation we get, 02+Z12+Z22=0+Z1Z2+0 ⇒Z12+Z22=Z1Z2 ⇒Z12+Z22=b[. as Z1Z2=b] ⇒(Z1+Z2)2−2Z1Z2=b ⇒(Z1+Z2)2−2b=b ⇒(Z1+Z2)2=3b ⇒(−a)2=3b ⇒a2=3b So Option (D) is correct. [ Note : This question is asked to check if you know the following formula - "If Z1,Z2 and Z3 are the vertices of an equilateral triangle then - Z12+Z22+Z32=Z1Z2+Z2Z3+Z3Z1′′]