2021 ACJC Promo Q8

The figure below shows a cross-section \(OBCE\) of a car headlight whose reflective surface is modelled in suitable units by the curve with parametric equations

\(x=a\left( \theta -\sin \theta \right),y=a\left( 1-\cos \theta \right)\)

for \(0\le \theta \le 2\pi \), where \(a\) is a positive constant.

1. Find in terms of \(a\)
   1. the length of \(OE\),[2]
   2. the maximum height of the curve \(OBCE\).[1]
2. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x}=\cot \frac{\theta }{2}\).[3]

Point \(B\) lies on the curve and has parameter \(\beta \). \(TS\) is tangential to the curve at \(B\) and \(BC\) is parallel to the \(x\)-axis.

Given that \(\angle TBC=\frac{\pi }{6}\),

3. Show that \(\beta =\frac{2\pi }{3}\).[2]
4. Show that the equation of normal to the curve at the point \(B\) is
   \(ky=-{{k}^{2}}x+2\pi a\),
   where \(k\) is an exact constant to be determined.[3]