1.3: A rational curve and its normal

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 calculus, show that at every point on the curve, the gradient of $\displaystyle C$ is negative.

(d) Using the axes on the opposite page, sketch $\displaystyle C$. Show clearly and label with their equation any asymptotes and the coordinates of the points of intersection of $\displaystyle C$ with the coordinate axes.

The straight line $\displaystyle l$ is the normal to $\displaystyle C$ at the point $\displaystyle A$. The $\displaystyle x$ coordinate of $\displaystyle A$ is positive and the gradient of $\displaystyle l$ is $\displaystyle 17$. The line $\displaystyle l$ also intersects $\displaystyle C$ at the point $\displaystyle B$.

(e) Find the exact coordinates of $\displaystyle B$.