1.4: Asymptotes and intercepts of a rational curve

A curve $\displaystyle C$ has equation

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

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

(a) Show that $\displaystyle a=3$.

Hence

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

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

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

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

(i) the $\displaystyle x$-axis,

(ii) the $\displaystyle y$-axis.

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