Question

If the point $z_1$ is the reflection of a point $z_2$ through the line $\overline{b}z+b\overline{z}=c$, where $b\ne 0$, then $\overline{b}{z}_1+b\overline{z_2}$ is

• A. Proportional to $|z_1{|}^2-|z_2{|}^2$
• B. Equal to $c$
• C. Inversely proportional to $c$
• D. None of these

Answer 1

Answer: (A), (B)

Equal to $c$
Proportional to $|z_1{|}^2-|z_2{|}^2$

The given line is the perpendicular bisector of the line joining the 2 given points say A (z1) and B (z2).
Let R (z) be the intersection point of the 2 lines.
Thus, AR = BR => $|z-z1|=|z-z2|=>{|z-z1|}^2={|z-z2|}^2=>z(\overline{z1}-\overline{z2})+\overline{z}(z1-z2)={|z1|}^2-{|z2|}^2$
This is identical to $\overline{b}z+\overline{z}b=c$.
Hence, $b=z1-z2$ and $c={|z1|}^2-{|z2|}^2$
Thus, $\overline{b}z1+b\overline{z2}=(\overline{z1}-\overline{z2})z1+(z1-z2)\overline{z2}={|z1|}^2-{|z2|}^2=c$
Hence, (a), (b) are correct.