Question

If $z_1$ and $z_2$ lie on the same side the line $\overline{a}z+z\overline{z}+b=0$, where $a\in C,a\ne 0,b\in R$, then the ratio $\frac{\overline{a}{z}_1+a\overline{z_1}+b}{\overline{a}{z}_2+a\overline{z_2}+b}$ is

• A. Purely imaginary
• B. A positive real number
• C. A negative real number
• D. None of these

Answer 1

Answer: (B)

A positive real number

$z_1,z_2$ lie on same side of the line

Then $m=\frac{\overline{a}{z}_1+a\overline{z_1}+b}{\overline{a}{z}_2+a\overline{z_2}+b}$ is positive real number

If $z_1,z_2$ lie on the opposite sides

$m<0$