Year: 2007
Paper: 3
Question Number: 4
Course: LFM Pure and Mechanics
Section: Parametric equations
No solution available for this problem.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
A curve is given parametrically by
\begin{align*}
x&= a\big( \cos t +\ln \tan \tfrac12 t\big)\,,\\
y&= a\sin t\,,
\end{align*}
where $0 < t < \frac12 \pi$ and $a$ is a positive constant. Show that
$\ds \frac{\d y}{\d x} = \tan t$ and sketch the curve.
Let $P$ be the point with parameter $t$ and let $Q$ be the
point where the tangent
to the curve at $P$ meets the $x$-axis.
Show that $PQ=a$.
The {\sl radius of curvature}, $\rho$, at $P$
is defined by
\[
\rho=
\frac
{\big(\dot x ^2+\dot y^2\big)^{\frac32}}
{\vert \dot x \ddot y - \dot y \ddot x\vert \ \ }
\,,
\]
where the dots denote differentiation with respect to $t$. Show that
$\rho =a\cot t$.
The point $C$
lies on the normal to the curve at $P$, a distance $\rho$ from $P$
and above the curve.
Show that $CQ$ is parallel to the $y$-axis.
This question was quite popular. A lot of attempts involved rambling trigonometrical manipulations, and few spotted the standard differential of ln tan t/2. The curve sketch was often omitted or incorrect, and there was a lot of complicated working using e.g. the equation of the normal etc. to find the centre of curvature.