Question

One vertex of an equilateral triangle is at the origin and the other two vertices and given by $2z^2+2z+k=0$, then $k$ is

• A. $=1$
• B. $<\frac{1}{2}$
• C. $>\frac{1}{2}$
• D. $<0$

Answer 1

Answer: (C)

$>\frac{1}{2}$

If origin and complex numbers a and b makes an equilateral triangle, they are related as:

$a^2$ + $b^2$ = $a\times b$ - (1)

For equation $2z^2+2z+k=0$, we know that

I) sum of roots $=-1$

ii) product of roots $=k/2$

From (1),

$a^2$ + $b^2$ = $a\times b$

$\Rightarrow$ $(a+b{)}^2-2ab=a\times b$

$\Rightarrow$ $(-1{)}^2-2(k/2)=k/2$

$\Rightarrow k=2/3>1/2$

Hence, the correct option is C.