<p>The parametric equations of the curve \(C\) are</p>
<p style="text-align: center">\[x = 1 - 3\csc \theta \text{ and } y = 2\cot \theta - 3, \text{ where } 0 \leq \theta \leq \pi.\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show \(\frac{\mathrm{d}y}{\mathrm{d}x} = -\frac{2}{3}\sec \theta\). Hence find the equation of the normal to \(C\) at the point where \(\theta = \frac{\pi}{4}\). Give the equation in the form \(y = Ax + B\), where \(A\) and \(B\) are exact constants to be found.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the normal found in part (a) will cut \(C\) again.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the Cartesian equation of \(C\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch \(C\), indicating clearly its key features.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the range of values of \(m\) such that there is no intersection between the line \(y = m(x - 1) - 3\) and \(C\).</span><span class="inline-marks">[2]</span></div></li></ol>

The parametric equations of the curve \(C\) are

\[x = 1 - 3\csc \theta \text{ and } y = 2\cot \theta - 3, \text{ where } 0 \leq \theta \leq \pi.\]

Show \(\frac{\mathrm{d}y}{\mathrm{d}x} = -\frac{2}{3}\sec \theta\). Hence find the equation of the normal to \(C\) at the point where \(\theta = \frac{\pi}{4}\). Give the equation in the form \(y = Ax + B\), where \(A\) and \(B\) are exact constants to be found.

Show that the normal found in part (a) will cut \(C\) again.

Sketch \(C\), indicating clearly its key features.

Find the range of values of \(m\) such that there is no intersection between the line \(y = m(x - 1) - 3\) and \(C\).

2025 HCI P2 Q3

2025 HCI P2 Q3