1.7: Differentiate and sketch a rational curve

A curve $\displaystyle C$ has equation

where $\displaystyle a$ is a constant and $\displaystyle x\neq\frac{6}{a}$.

(a) Find $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$.

The curve crosses the $\displaystyle y$-axis at the point $\displaystyle A$.

The normal to $\displaystyle C$ at the point $\displaystyle A$ is the line $\displaystyle l$ with equation

Show that

(b) (i) $\displaystyle a=3$,

(ii) the equation of $\displaystyle C$ is

(c) Using the axes on the opposite page, sketch $\displaystyle C$, showing clearly the asymptotes with their equations and the coordinates of the points where $\displaystyle C$ crosses the coordinate axes.

The line $\displaystyle l$ meets $\displaystyle C$ again at the point $\displaystyle D$.

(d) Find the $\displaystyle x$ coordinate of $\displaystyle D$.

Give your answer as an improper fraction.