2016 ACJC MYE Q4

The curve \(C\) has parametric equation \(x=3-2\cos \,t\), \(y=-\tan \,t\) for \(0 < t < \frac{\pi }{2}\). Using differentiation, show that the curve \(C\) does not have any stationary point.[2]

1. A point \(Q\) on \(C\) has parameter \(t=\frac{\pi }{6}\). Find the equation of the tangent at \(Q\), leaving your answer in exact form.[2]
2. Find the cartesian equation of the locus of the mid-point of \((3-2\cos \,t,\,-\tan \,t)\) and \((-3,0)\) as \(t\) varies.[3]