Question

Consider $a{z}^2+bz+c=0$, where $a,b,c\in R$ and $4ac>b^2$
In the argand's plane. if A is the point represnting $z_1$. B is the point representing $z_2$ and $z=\frac{\overrightarrow{OA}}{\overrightarrow{OB}}$ then z is:

• A. z is purely real
• B. z is purely imaginary
• C. $\left|z\right|=1$
• D. $\Delta AOB$ is a scalene triangle

Answer 1

Answer: (A)

z is purely real

Let $Z_1=a{c}^{iq},Z_2=a{c}^{-iq}$

Since $Z_1$ and $Z_2$ are conjugate points,

$Z=\frac{\overline{OA}}{\overline{OB}}=\frac{a{c}^{iq}}{a{c}^{-iq}}=c^{2iq}$

So, $|Z|=1.$