Question

$z$ is a complex number. Origin and the roots of $z^2+pz+q=0$ form an equilateral triangle, if

• A. $p^2=q$
• B. $p^2=3q$
• C. $q^2=p$
• D. $q^2=3p$

Answer 1

The correct option is B $p^2=3q$

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 $z^2+pz+q=0$, we know that

I) sum of roots $=-p$

ii) product of roots $=q$

Now if 'a' and 'b' are roots of $z^2+pz+q=0$, then

$a+b=-p$

$a\times b=q$

From (1),

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

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

$\Rightarrow$ $(-p{)}^2-2q=q$

$\Rightarrow$ $p^2=3q$

Hence, option B is correct.