The curve C has parametric equations : $x = \sin \theta + 1$ and $y = \sqrt{3} \cos \theta - 1$,

Question

The curve C has parametric equations : $x = \sin \theta + 1$ and $y = \sqrt{3} \cos \theta - 1$, where $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$ Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = -\sqrt{3} \tan \theta$.[2] The tangent to $C$ at point $T(\sin t + 1, \sqrt{3} \cos t - 1)$ makes an angle of $\frac{3\pi}{4}$ with the positive $x$-axis. Find the equation of the normal to $C$ at $T$.[3] Find the exact $y$-coordinate of the point on $C$ where the tangent to $C$ is parallel to the $x$-axis.[2] Find the area of the quadrilateral bounded by the axes, the normal to $C$ at $T$ and the tangent in (c).[3]

Accepted Answer

(a) $\frac{\mathrm{d}y}{\mathrm{d}x}=-\sqrt3\tan\theta$ (shown). (b) $y=x-1$. (c) $\sqrt3-1$. (d) $1$ square unit.

2025 JPJC P2 Q3

2025 JPJC P2 Q3

Solution:

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