Question

Reflection of the line $\overline{a}z+a\overline{z}=0$ in the real axis is

• A. $\overline{az}+az=0$
• B. $\overline{\frac{a}{a}}+\frac{Z}{Z}=0$
• C. $(a+\overline{a})(z+\overline{z})=0$
• D. az $=0$

Answer 1

The correct option is A $\overline{az}+az=0$

Let $a=\alpha +i\beta$ $z=x+iy$
Now,
$\overline{a}z+a\overline{z}=0$
$\Rightarrow (\alpha -i\beta )(x+iy)+(\alpha +i\beta )(x-iy)=0$
$\Rightarrow 2(\alpha x+\beta y)=0\Rightarrow \alpha x+\beta y=0\to$ line passes through origin.
Slope $=\frac{-\alpha x}{\beta }$
So reflection slope $=\frac{\alpha }{\beta }x$
$\Rightarrow$ line is $\alpha x-\beta y=0\to$reflection also passes through origin
$\Rightarrow \left(\frac{a+\overline{a}}{2}\right)\left(\frac{z+\overline{z}}{2}\right)-\left(\frac{a-\overline{a}}{2i}\right)\left(\frac{z-\overline{z}}{2i}\right)=0$
$\Rightarrow az+\overline{a}\overline{z}=0$