5 This question concerns curves with polar equation \(r = \sec \theta + a\), where \(a\) is a constant.
- State the set of values of \(\theta\) between 0 and \(2 \pi\) for which \(r\) is undefined.
For the rest of the question you should assume that \(\theta\) takes all values between 0 and \(2 \pi\) for which \(r\) is defined.
- Use your graphical calculator to obtain a sketch of the curve in the case \(a = 0\). Confirm the shape of the curve by writing the equation in cartesian form.
- Sketch the curve in the case \(a = 1\).
Now consider the curve in the case \(a = - 1\). What do you notice?
By considering both curves for \(0 < \theta < \pi\) and \(\pi < \theta < 2 \pi\) separately, describe the relationship between the cases \(a = 1\) and \(a = - 1\). - What feature does the curve exhibit for values of \(a\) greater than 1 ?
Sketch a typical case.
- Show that a cartesian equation of the curve \(r = \sec \theta + a\) is \(\left( x ^ { 2 } + y ^ { 2 } \right) ( x - 1 ) ^ { 2 } = a ^ { 2 } x ^ { 2 }\).