<p>The curve \(C_1\) has equation</p>
<p style="text-align: center">\((x - 1)^2 + 4y^2 = 4.\)</p>
<p>The curve \(C_2\) has parametric equations</p>
<p style="text-align: center">\(x = \sec\theta \text{ and}\hspace{0.5em} y = 2\tan\theta.\)</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that the cartesian equation of \(C_2\) is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where&nbsp;\(a\) and \(b\) are positive integers to be determined.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Sketch \(C_1\) and \(C_2\) on the same diagram, indicating clearly the equations of any asymptotes and coordinates of vertices.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;\(k &gt; 0\), find the range of values of \(k\) such that&nbsp;\(k(x - 1)^2 + 4y^2 = 4\) cuts \(C_2\) at most twice.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Describe a sequence of transformations that maps \(C_1\) to a unit circle with centre \((0, 0)\).<span class="inline-marks">[2]</span></div></li></ol>

The curve \(C_2\) has parametric equations

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

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.

Sketch \(C_1\) and \(C_2\) on the same diagram, indicating clearly the equations of any asymptotes and coordinates of vertices.

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.

Describe a sequence of transformations that maps \(C_1\) to a unit circle with centre \((0, 0)\).

2024 VJC Promo Q8

2024 VJC Promo Q8