How to show that $|(1+ia)/(1-i/a)| = a$?

Question

I'm struggling to show, that:

$$\Bigg| \dfrac{1+\mathrm{i}\cdot a}{1+\mathrm{i}/a} \Bigg| = a \;.$$

($\mathrm{i} $ denoting the imaginary unit, $a$ is a real, postive number)

Is there some simple trick I'm missing or any theorem helping to show, that this equation is true?

Answer 1

Hint. Note that if $a$ is a positive real number then $$|1+ia|=\sqrt{1+a^2}\quad\text{and}\quad |1+i/a|=\sqrt{1+1/a^2}.$$ Moreover, recall that $|z/w|=|z|/|w|$ for any $z,w\in \mathbb{C}$ with $w\neq 0$.

Answer 2

Note that $$1+i/a=i/a\,(1-ia)$$ and $$ |1+ia|=|1-ia|. $$

Answer 3

$$\left|\frac{1+ia}{1+\frac ia}\right|=\frac{|1+ia|}{\left|1+\frac ia\right|} = \frac{\sqrt{1^2+a^2}}{\sqrt{1+\left(\frac1a\right)^2}} = \sqrt{\frac{1^2+a^2}{1+\left(\frac1a\right)^2}}$$

can you continue from here?