1.18: Past-paper question 10

A curve $\displaystyle C$ has equation

(a) Find the coordinates of the point where $\displaystyle C$ intersects the

(i) $\displaystyle x$-axis

(ii) $\displaystyle y$-axis

(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) Sketch $\displaystyle C$ on the opposite page.

Show and label the asymptotes and the coordinates of the points where $\displaystyle C$ crosses the coordinate axes.

Point $\displaystyle A$ lies on $\displaystyle C$ such that the gradient of $\displaystyle C$ at $\displaystyle A$ is parallel to the line with equation $\displaystyle 4y - x = 7$

The normal to $\displaystyle C$ at $\displaystyle A$ intersects the $\displaystyle x$-axis at point $\displaystyle D$ and the $\displaystyle y$-axis at point $\displaystyle E$
Given that the $\displaystyle x$ coordinate of $\displaystyle A$ is positive,

(d) find, in its simplified form, the exact length of line $\displaystyle DE$