5 The points \(\mathrm { A } ( - 1,0 ) , \mathrm { B } ( 1,0 )\) and \(\mathrm { P } ( x , y )\) are such that the product of the distances PA and PB is 1 . You are given that the cartesian equation of the locus of P is $$\left( ( x + 1 ) ^ { 2 } + y ^ { 2 } \right) \left( ( x - 1 ) ^ { 2 } + y ^ { 2 } \right) = 1 .$$

1. Show that this equation may be written in polar form as $$r ^ { 4 } + 2 r ^ { 2 } = 4 r ^ { 2 } \cos ^ { 2 } \theta$$ Show that the polar equation simplifies to $$r ^ { 2 } = 2 \cos 2 \theta$$
2. Give a sketch of the curve, stating the values of \(\theta\) for which the curve is defined.
3. The equation in part (i) is now to be generalised to $$r ^ { 2 } = 2 \cos 2 \theta + k$$ where \(k\) is a constant.
   (A) Give sketches of the curve in the cases \(k = 1 , k = 2\). Describe how these two curves differ at the pole.
   (B) Give a sketch of the curve in the case \(k = 4\). What happens to the shape of the curve as \(k\) tends to infinity?
4. Sketch the curve for the case \(k = - 1\). What happens to the curve as \(k \rightarrow - 2\) ? \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}