1.21: Past-paper question 10

A curve $\displaystyle C$ has equation

(a) Write down an equation of the asymptote to $\displaystyle C$ that is

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

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

(b) Find the coordinates of the points of intersection of $\displaystyle C$ with the coordinate axes.

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

$\displaystyle C$ passes through the point $\displaystyle A$ with coordinates $\displaystyle \left(\frac{1}{2}, \frac{13}{8}\right)$

(d) Show that the gradient of $\displaystyle C$ at $\displaystyle A$ is $\displaystyle -\frac{17}{16}$

$\displaystyle C$ also passes through the point $\displaystyle P$

Given that the tangent to $\displaystyle C$ at $\displaystyle P$ is parallel to the tangent to $\displaystyle C$ at $\displaystyle A$

(e) find an equation of the tangent to $\displaystyle C$ at $\displaystyle P$

Give your answer in the form $\displaystyle ax + by + c = 0$ where $\displaystyle a, b$ and $\displaystyle c$ are integers.