5 This question concerns the curves with polar equation $$r = \sec \theta + a \cos \theta ,$$ where \(a\) is a constant which may take any real value, and \(0 \leqslant \theta \leqslant 2 \pi\).

1. On a single diagram, sketch the curves for \(a = 0 , a = 1 , a = 2\).
2. On a single diagram, sketch the curves for \(a = 0 , a = - 1 , a = - 2\).
3. Identify a feature that the curves for \(a = 1 , a = 2 , a = - 1 , a = - 2\) share.
4. Name a distinctive feature of the curve for \(a = - 1\), and a different distinctive feature of the curve for \(a = - 2\).
5. Show that, in cartesian coordinates, equation (*) may be written $$y ^ { 2 } = \frac { a x ^ { 2 } } { x - 1 } - x ^ { 2 }$$ Hence comment further on the feature you identified in part (iii).
6. Show algebraically that, when \(a > 0\), the curve exists for \(1 < x < 1 + a\). Find the set of values of \(x\) for which the curve exists when \(a < 0\).