2025 HCI P2 Q3
2025 HCI P2 Q3
13 marks
The parametric equations of the curve \(C\) are
\[x = 1 - 3\csc \theta \text{ and}\hspace{0.5em} y = 2\cot \theta - 3, \text{ where}\hspace{0.5em} 0 \leq \theta \leq \pi.\]
- 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.[4]
- Show that the normal found in part (a) will cut \(C\) again.[2]
- Find the Cartesian equation of \(C\).[2]
- Sketch \(C\), indicating clearly its key features.[3]
- Find the range of values of \(m\) such that there is no intersection between the line \(y = m(x - 1) - 3\) and \(C\).[2]
Solution:
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Answer:(a) \(\frac{\mathrm dy}{\mathrm dx}=-\frac23\sec\theta\); normal \(y=\frac{3\sqrt2}4x+\frac72-\frac{3\sqrt2}4\). (b) Second intersection shown, at \(\theta\approx2.95319\). (c) \(\frac{(x-1)^2}9-\frac{(y+3)^2}4=1\), \(x\le-2\). (d) Left-branch sketch shown. (e) \(m\le-\frac23\) or \(m\ge\frac23\).
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