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{\bf Question}

transform the following equations into cartesian co-ordinates and
identify the curves they represent.

\begin{description}
\item[(i)] $\ds r^2 \cos \theta = 1$
\item[(ii)] $\ds r^2 \sin 2 \theta = 1$
\end{description}

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{\bf Answer}

\begin{description}
\item[(i)] $\ds r^2 \cos \theta = 1$ so $\ds r^2(\cos^2\theta - \sin^2
\theta) = 1$

Thus in cartesian $\ds x^2 + y^2=1$ which is a rectangular
hyperbola.
\item[(ii)] $\ds r^2 \sin 2 \theta = 1 \Rightarrow 2r^2 \sin \theta \cos \theta = 1$

Thus in cartesian $\ds 2xy=1$ which is a rectangular hyperbola.
\end{description}




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