1.23: Past-paper question 10

The curve $\displaystyle C$ has equation

The asymptote to $\displaystyle C$ that is parallel to the $\displaystyle y$-axis has equation $\displaystyle x = -\frac{4}{3}$

$\displaystyle C$ crosses the $\displaystyle y$-axis at the point with coordinates $\displaystyle \left(0, -\frac{1}{4}\right)$

(a) (i) Show that $\displaystyle b = 8$

(ii) Find the value of $\displaystyle a$

(b) Write down the equation of the asymptote to $\displaystyle C$ that is parallel to the $\displaystyle x$-axis.

(c) Find the coordinates of the point where $\displaystyle C$ crosses the $\displaystyle x$-axis.

(d) Using calculus, show that at every point on $\displaystyle C$, the gradient is positive.

(e) Using the axes on the next page, sketch $\displaystyle C$

Clearly label the asymptotes and the coordinates where $\displaystyle C$ crosses the coordinate axes.

The gradient of $\displaystyle C$ is $\displaystyle \frac{13}{100}$ at the point $\displaystyle E$ and at the point $\displaystyle F$

(f) Find the length of the line $\displaystyle EF$, giving your answer to 2 decimal places.