Question

Let $z_1$ and $z_2$ be roots of the equation $z^2+pz+q=0,$ where p and q may be complex numbers. Let A and B represents $z_1$ and
$z_2$ in the complex plane. Given
$\angle AOB=\alpha \ne 0$ and $OA=OB$ where O is the origin. Using the given information what will be the value of $p^2?$

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

Since $\overline{OB}$ is obtained by rotating $\overline{OA}$ through $\alpha ,$,hence$\overline{OB}=\overline{OA}{e}^{i\alpha }$

$\Rightarrow z_2-0=\left(z_{1}0\right)e^{i\alpha }\Rightarrow \frac{z_2}{z_1}=cos\alpha +isin\alpha$

$\Rightarrow \frac{z_2}{z_1}=2co{s}^2\frac{\alpha }{2}-1+2isin\frac{\alpha }{2}=cos\frac{\alpha }{2}$

$\Rightarrow \frac{z_1+z_2}{z_1}=2cos\frac{\alpha }{2}(cos\frac{\alpha }{2}+isin\frac{\alpha }{2})$

$\Rightarrow \frac{(z_1+z_2{)}^2}{z_1^2}=4co{s}^2\frac{\alpha }{2}{(cos\frac{\alpha }{2}+isin\frac{\alpha }{2})}^2$

$\Rightarrow 4co{s}^2\frac{\alpha }{2}(cos\alpha +isin\alpha )=4co{s}^2\frac{z_2}{z_1}$[from(i)]

$\Rightarrow (z_1+z_2{)}^2=4co{s}^2\frac{\alpha }{2}\left(z_2{z}_1\right)\Rightarrow (-p{)}^2=4co{s}^2\frac{\alpha }{2}\left(q\right)$

$(\because z_{1}and{z}_2$ and the roots of $z^2+pz+q=0)$
Therefore,$p^2=4qco{s}^2\frac{\alpha }{2}$ $\because z_1+z_2=-p,z_1{z}_2=q).$