1.11: Reciprocal curves and asymptotes and curve intersections

The curve $C$ has equation $y = \frac{ax - 5}{b - x}$ where $a$ and $b$ are integers and $x \neq b$
One intersection of $C$ with the coordinate axes is at the point with coordinates $\left(\frac{5}{4}, 0\right)$

The asymptote parallel to the $y$-axis has equation $x = 3$

(a) Find the value of $a$ and the value of $b$

(b) Sketch $C$, showing clearly the asymptotes with their equations and the coordinates of the points of intersection with the coordinate axes.

The straight line $l$ with equation $4y - 7x = k$ has no points of intersection with $C$

(c) Show, using algebra, that the range of possible values of $k$ can be written as

where $m$ and $n$ are integers to be found.