Study of a homographic function with its inverse

Text

Given the homographic function $y = \dfrac{2x + 3}{x - 1}$:

• (a) find centre and asymptotes;
• (b) write it in implicit form $(x-x_c)(y-y_c)=k$;
• (c) find vertices and foci;
• (d) find the equation of the inverse.

Solution

(a) Vertical asymptote $x=1$; horizontal $y=2$. Centre $C(1;2)$.

(b) $(x-1)(y-2)=5$, so $k=5>0$.

(c) Rectangular hyperbola: vertices on the bisector $y-2=x-1$, with $(x-1)^2=5$, i.e. $V\left(1\pm\sqrt5;2\pm\sqrt5\right)$. Semi-axes $a=b=\sqrt{10}$, $c=2\sqrt5$; foci $F\left(1\pm\sqrt{10};2\pm\sqrt{10}\right)$.

(d) Swapping $x\leftrightarrow y$: $y=\dfrac{x+3}{x-2}$.

$\boxed{C(1;2);\ (x-1)(y-2)=5;\ y_{inv}=\tfrac{x+3}{x-2}}$