Show that $|{|2\overline{z}+5 |(\sqrt2 - i)} | = \sqrt3 |2z+5|$, where z is a complex number.

Question

Show that $|{|2\overline{z}+5 |(\sqrt2 - i)} | = \sqrt3 |2z+5|$, where z is a complex number.

And $\overline{z}$ is complex conjugate of $z$.

And $i$ is iota.

I'm proceeding by considering $z=x+iy$

But I just get stuck at different results approaching different ways. Please help.

Answer 1

LHS is $||2\overline{z}+5|\cdot(\sqrt2 - i)|$; and $a=|2\overline{z}+5| \geq 0$. Then $$||2\overline{z}+5|\cdot(\sqrt2 - i)|=|a\cdot(\sqrt2 - i)|=|a|\cdot|\sqrt2 - i|\\=\sqrt3 \cdot a=\sqrt3 \cdot |2\overline{z}+5|=\sqrt3 \cdot |\overline{2z+5}|$$