2024 VJC Promo Q8

The curve \(C_1\) has equation

\((x - 1)^2 + 4y^2 = 4.\)

The curve \(C_2\) has parametric equations

\(x = \sec\theta \text{ and}\hspace{0.5em} y = 2\tan\theta.\)

1. Show that the cartesian equation of \(C_2\) is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are positive integers to be determined.[2]
2. Sketch \(C_1\) and \(C_2\) on the same diagram, indicating clearly the equations of any asymptotes and coordinates of vertices.[4]
3. Given that \(k > 0\), find the range of values of \(k\) such that \(k(x - 1)^2 + 4y^2 = 4\) cuts \(C_2\) at most twice.[2]
4. Describe a sequence of transformations that maps \(C_1\) to a unit circle with centre \((0, 0)\).[2]