2025 ACJC Promo Q5

A curve \(C_1\) has parametric equation

1. Sketch the curve \(C_1\), stating the coordinates of the endpoint(s).[1]
2. Show that \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{1}{3\tan\theta}\).[2]

\(L_1\) and \(L_2\) are tangents to curve \(C_1\).

3. Find
   1. the equation of \(L_1\) which is parallel to the \(y\)-axis.[2]
   2. the gradient of \(L_2\) which is tangent to the curve \(C_1\) at the point \((3.75,1)\).[2]
4. \(\beta\) is the acute angle between \(L_1\) and \(L_2\). State the value of \(\tan\beta\).[1]
5. Describe a transformation that will map \(C_1\) onto the curve \(C_2\) with equations \(x=3\sec^2\theta\), \(y=2\tan(\pi-\theta)\), where \(0\leq\theta<\dfrac{\pi}{2}\).[1]