A complex conjugation mulplication

Question

Compute ||a|| given that e/ā=5/4+i/4 and ea=10+2i

I found sqrt(13) as answer, but the solution says its 2sqrt(2), am I doing something wrong?

Thanks a lot!

Answer 1

$$e\cdot a=10+2i$$

$$\frac e {\bar a} =\frac{5+i}4$$

On division $$a\cdot \bar a=\frac{4(10+2i)}{5+i}=8\cdot \frac{5+i}{5+i}=8$$

If $a=x+iy, \bar a=x-iy,$ so, $a\cdot \bar a=(x+iy)(x-iy)=x^2+y^2=\mid a\mid ^2$

So, $$\mid a\mid ^2=8\implies |a|=2\sqrt2 $$

As $$e\cdot a=10+2i, \mid e\cdot a \mid=\sqrt{10^2+2^2}=\sqrt {104}$$

As $\mid e\cdot a \mid= \mid e\mid\cdot \mid a \mid, $

so, $\mid e\mid\cdot \mid a \mid= \sqrt {104}$

but, $\mid a \mid=\sqrt 8,$ so, $\mid e\mid=\frac{\sqrt{104}}{\sqrt{8}}=\sqrt{13}$