Challenging +1.2 This is a multi-part Further Maths question on polar coordinates requiring standard conversions (x=r cos θ, y=r sin θ), finding minima by differentiation, solving simultaneous polar equations, and applying the cosine rule. While it involves several steps and Further Maths content (inherently harder), each individual technique is routine for FP3 students with no novel insights required—making it moderately above average difficulty.
A curve has cartesian equation \(x y = 8\). Show that the polar equation of the curve is \(r ^ { 2 } = 16 \operatorname { cosec } 2 \theta\).
The diagram shows a sketch of the curve, \(C\), whose polar equation is
$$r ^ { 2 } = 16 \operatorname { cosec } 2 \theta , \quad 0 < \theta < \frac { \pi } { 2 }$$
\includegraphics[max width=\textwidth, alt={}, center]{c4bce668-61f1-4be0-97ee-c635df7e1fc6-4_364_567_1635_726}
Find the polar coordinates of the point \(N\) which lies on the curve \(C\) and is closest to the pole \(O\).
The circle whose polar equation is \(r = 4 \sqrt { 2 }\) intersects the curve \(C\) at the points \(P\) and \(Q\). Find, in an exact form, the polar coordinates of \(P\) and \(Q\).
The obtuse angle \(P N Q\) is \(\alpha\) radians. Find the value of \(\alpha\), giving your answer to three significant figures.
(5 marks)
8
\begin{enumerate}[label=(\alph*)]
\item A curve has cartesian equation $x y = 8$. Show that the polar equation of the curve is $r ^ { 2 } = 16 \operatorname { cosec } 2 \theta$.
\item The diagram shows a sketch of the curve, $C$, whose polar equation is
$$r ^ { 2 } = 16 \operatorname { cosec } 2 \theta , \quad 0 < \theta < \frac { \pi } { 2 }$$
\includegraphics[max width=\textwidth, alt={}, center]{c4bce668-61f1-4be0-97ee-c635df7e1fc6-4_364_567_1635_726}
\begin{enumerate}[label=(\roman*)]
\item Find the polar coordinates of the point $N$ which lies on the curve $C$ and is closest to the pole $O$.
\item The circle whose polar equation is $r = 4 \sqrt { 2 }$ intersects the curve $C$ at the points $P$ and $Q$. Find, in an exact form, the polar coordinates of $P$ and $Q$.
\item The obtuse angle $P N Q$ is $\alpha$ radians. Find the value of $\alpha$, giving your answer to three significant figures.\\
(5 marks)
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2012 Q8 [14]}}