5 A curve has parametric equations $x = \lambda \cos \theta - \frac { 1 } { \lambda } \sin \theta , y = \cos \theta + \sin \theta$, where $\lambda$ is a positive constant.\\
(i) Use your calculator to obtain a sketch of the curve in each of the cases

$$\lambda = 0.5 , \quad \lambda = 3 \quad \text { and } \quad \lambda = 5 .$$

(ii) Given that the curve is a conic, name the type of conic.\\
(iii) Show that $y$ has a maximum value of $\sqrt { 2 }$ when $\theta = \frac { 1 } { 4 } \pi$.\\
(iv) Show that $x ^ { 2 } + y ^ { 2 } = \left( 1 + \lambda ^ { 2 } \right) + \left( \frac { 1 } { \lambda ^ { 2 } } - \lambda ^ { 2 } \right) \sin ^ { 2 } \theta$, and deduce that the distance from the origin of any point on the curve is between $\sqrt { 1 + \frac { 1 } { \lambda ^ { 2 } } }$ and $\sqrt { 1 + \lambda ^ { 2 } }$.\\
(v) For the case $\lambda = 1$, show that the curve is a circle, and find its radius.\\
(vi) For the case $\lambda = 2$, draw a sketch of the curve, and label the points $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G } , \mathrm { H }$ on the curve corresponding to $\theta = 0 , \frac { 1 } { 4 } \pi , \frac { 1 } { 2 } \pi , \frac { 3 } { 4 } \pi , \pi , \frac { 5 } { 4 } \pi , \frac { 3 } { 2 } \pi , \frac { 7 } { 4 } \pi$ respectively. You should make clear what is special about each of these points.

\hfill \mbox{\textit{OCR MEI FP2 2008 Q5 [18]}}