Question

Let $z_1$ and $z_2$ be roots of the equation $z^2+pz+q=0$ where the coefficients $p$ and $q$ may be complex numbers. Let $A$ and $B$ represent $z_1$ and $z_2$ in the complex plane, If $\angle AOB=\theta \ne 0$ and $OA=OB$ where $O$ is the origin, then $p^2$ is

• A. $q{\cos }^2\left(\theta /2\right)$
• B. $2q{\cos }^2\left(\theta /2\right)$
• C. $3q{\cos }^2\left(\theta /2\right)$
• D. $4q{\cos }^2\left(\theta /2\right)$

Answer 1

The correct option is D $4q{\cos }^2\left(\theta /2\right)$


$\because z_1$ and $z_2$ be roots of the equation $z^2+pz+q=0$ Then ,
$\therefore z_1+z_2=-p,z_1{z}_2=q$
Also,
$\frac{z_2}{z_1}=e^{i\theta }\Rightarrow z_2=z_1{e}^{i\theta }$
$\Rightarrow z_1(1+e^{i\theta })=-p$
Now
$q=z_1^2{e}^{i\theta }$
$\Rightarrow \frac{p^2}{(1+e^{i\theta }{)}^2}=q{e}^{-i\theta }$
$\Rightarrow p^2=q{e}^{-i\theta }(1+e^{2i\theta }+2e^{i\theta })$
$=q(e^{-i\theta }+e^{i\theta }+2)$
$=q(2\cos \theta +2)$
$=4q{\cos }^2\frac{\theta }{2}$