1.15: Past-paper question 10

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

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

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

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

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

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

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