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, respectively. If $\angle AOB=\theta \ne 0$ and OA = OB, where O is the origin, then $p^2=4qco{s}^2\left(\theta /2\right)$. If this is true enter1, else enter 0.

Answer 1

From the above para we can conclude
$|z_1|=|z_2|=a$ .... (distance from origin is same)
And
$arg\left(z_1\right)-arg\left(z_2\right)=\theta$
$arg\left(z_1\right)=\theta +arg\left(z_2\right)$
Hence
$z_1=a{e}^{i(\theta +arg(z_2)}$
$=a{e}^{i\arg \left(z_2\right)}.e^{i\theta }$
$=\left(a{e}^{i\arg \left(z_2\right)}\right).e^{i\theta }$
$=z_2.e^{i\theta }$
Now
$z_1.z_2$
$=z_2^2.e^{i\theta }=q$
and
$z_1+z_2$
$=-p$
$=z_2(1+e^{i\theta })$
$=z_2(1+cos\theta +isin\theta )$
$=z_{2}2cos\frac{\theta }{2}\left(e^{i\frac{\theta }{2}}\right)$
$=2z_{2}cos\frac{\theta }{2}\left(e^{i\frac{\theta }{2}}\right)$
Now
$p^2=4z_2^{2}co{s}^2\frac{\theta }{2}(e^{i\frac{\theta }{2}}{)}^2$
$=4z_2^{2}co{s}^2\frac{\theta }{2}\left(e^{i\theta }\right)$
$=z_2^2.\left(e^{i\theta }\right)\left(4co{s}^2\frac{\theta }{2}\right)$
$=q\left(4co{s}^2\frac{\theta }{2}\right)$
$=4qco{s}^2\frac{\theta }{2})$