Let Z1 and Z2 be two roots of the equation Z2+aZ+b=0, Z being complex. Further , assume that the origin, Z1 and Z2 form an equilateral triangle. Then :
Solutions · 1
Official solution
Given quadratic equation,
Z2+aZ+b=0
and two roots are Z1 and Z2.
\[\therefore\] Z1 + Z2 = −a and Z1Z2=b\[
Question says,
There are three complex numbers:
1. Origin (0)
2. \]{Z_1}3.{Z_2}\[
and they form an equilateral triangle. So They are the vertices of the triangle.
[ Important Point : If \]{Z_1},Z2and{Z_3}\[ are the vertices of an equilateral triangle then -
\]Z_1^2+Z22+Z32=Z1Z2+Z2Z3+{Z_3}{Z_1}\[ ]
In this question,
\]{Z_1}= 0,Z2=Z1andZ3={Z_2}\[
By putting those values in the equation we get,
\]{0^2}+Z12+Z22=0+Z1Z2+ 0⇒Z12+Z22=Z1Z2⇒Z12+Z22=b\[ [ as \]{Z_1}\[ {Z\_2}\] = b ]
\[ \Rightarrow \] \[{\left( {{Z_1} + {Z_2}} \right)^2}\] - 2Z1Z2 = b
\[ \Rightarrow \] \[{\left( {{Z_1} + {Z_2}} \right)^2}\] - 2b = b
\[ \Rightarrow \] \[{\left( {{Z_1} + {Z_2}} \right)^2}\] = 3b
\[ \Rightarrow \] \[{\left( { - a} \right)^2}\] = 3b
\[ \Rightarrow \] 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" ]