Magnitude of a complex number

Question

How might we show that $\Big|{b^2+d^2-a^2-c^2+i2ab+i2cd\over a^2+b^2+c^2+d^2+2}\Big|\le 1$ if we are given that $ad-bc=1$ and $a,b,c,d$ are real?

Answer 1

If you write $z = (ia + b)$ and $w = (ic + d)$, the denominator is $z^2 + w^2$ and the numerator is $|z|^2 + |w|^2 + 2$. Then you can use the triangle inequality.