A curve $C$ has parametric equations \[x = \tan\theta - \cot\theta, \quad y = \tan\theta + \cot\th

Question

A curve $C$ has parametric equations $$x = \tan\theta - \cot\theta, \quad y = \tan\theta + \cot\theta, \quad \text{for } 0 < \theta < \frac{\pi}{4}.$$ Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = -\cos 2\theta$.[3] The line $L$ is the tangent to $C$ at point $P$ with parameter $p$. Show that the equation of $L$ is $y + (\cos 2p)x = 2\sin 2p$.[3] Let $Q$ and $R$ be the points where $L$ cuts the $x$-axis and $y$-axis respectively. Given that $p$ increases at a constant rate of $0.2$ radians per second, find the exact rate of change of the area of $\Delta OQR$ when $p = \frac{\pi}{6}$, where $O$ is the origin.[4]

Accepted Answer

(i) $\frac{\mathrm dy}{\mathrm dx}=-\cos2\theta$ (shown). (ii) $y+(\cos2p)x=2\sin2p$ (shown). (iii) $2\sqrt3$ square units per second.

2025 SAJC P2 Q5

2025 SAJC P2 Q5

Solution:

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