1.21: Asymptotes and intercepts of a rational curve

A curve $C$ has equation

where $a$ is a constant and $x\neq-5$.

The gradient of $C$ at the point on the curve where $x=2$ is $\frac{18}{49}$.

(a) Show that $a=3$.

Hence

(b) write down an equation of the asymptote to $C$ that is

(i) parallel to the $x$-axis,

(ii) parallel to the $y$-axis,

(c) find the coordinates of the point where $C$ crosses

(i) the $x$-axis,

(ii) the $y$-axis.

(d) Sketch the curve $C$, showing clearly its asymptotes and the coordinates of the points where $C$ crosses the coordinate axes.