1.10: Past-paper question 10

The curve $\displaystyle C$ with equation $\displaystyle y = \frac{6 - 3x}{x - 4}$ where $\displaystyle x \neq 4$, crosses the $\displaystyle x$-axis at the point $\displaystyle P$ and the $\displaystyle y$-axis at the point $\displaystyle Q$

(a) Find the coordinates of

(i) $\displaystyle P$

(ii) $\displaystyle Q$

(b) Write down an equation of the asymptote to $\displaystyle C$ which is

(i) parallel to the $\displaystyle y$-axis

(ii) parallel to the $\displaystyle x$-axis

(c) Sketch $\displaystyle C$ showing clearly the asymptotes and the coordinates of the points $\displaystyle P$ and $\displaystyle Q$

The line $\displaystyle L$ is the normal to $\displaystyle C$ at the point on $\displaystyle C$ where $\displaystyle x = 2$

(d) Find an equation of $\displaystyle L$

The line $\displaystyle L$ intersects $\displaystyle C$ again at the point $\displaystyle R$

(e) Find the $\displaystyle x$ coordinate of $\displaystyle R$