Why does ratio of $2$ linear expression gives you a rectangular hyperbola

Question

We were studying Calculus and some methods on how to find domain and range, when my teacher suddenly said, "FYI, the ratio of $2$ linear expression gives you a rectangular hyperbola

Can someone tell why does that give a "rectangular hyperbola only?"

Answer 1

Observe that, $ y= \dfrac{ax + b}{cx+d} = \dfrac{\frac{a}{c} \left(cx + d \right) - \frac{ad}{c} + b}{cx+d} = \dfrac{a}{c} + \dfrac{bc-ad}{c} \cdot \dfrac{1}{cx+d}$

Setting $\dfrac{a}{c} = K $ and $\dfrac{bc-ad}{c}= A $, and so $y= K + \dfrac{A}{cx+d} $ which is basically of the form: $ y = \dfrac{A}{x}$ and this is the equation of a rectangular hyperbola whose asymptotes are the co-ordinate axes.

---

Footnote: Rectangular hyperbola of the form: $y=\frac{A}{x}$