A curve $C$ has parametric equations $x=a( 2\cos heta -\cos 2 heta)$, $y=a( 2\sin heta -\sin 2\

Question

A curve $C$ has parametric equations $x=a( 2\cos 	heta -\cos 2	heta)$, $y=a( 2\sin 	heta -\sin 2	heta)$, for $0\le 	heta \le 2\pi .$ Sketch $C$ and state the Cartesian equation of its line of symmetry. [2] Find the values of $	heta $ at the points where $C$ meets the $x$-axis. [2] Show that the area enclosed by the $x$-axis, and the part of $C$ above the $x$-axis, is given by $\displaystyle\int
olimits_{{{	heta }_{1}}}^{{{	heta }_{2}}}{{{a}^{2}}\left( 4{{\sin }^{2}}	heta -6\sin 	heta \sin 2	heta +2{{\sin }^{2}}2	heta ight)}\,\mathrm{d}	heta ,$ where ${{	heta }_{1}}$ and ${{	heta }_{2}}$ should be stated. [3] Hence find, in terms of $a,$ the exact total area enclosed by $C.$ [5]

Accepted Answer

(ii) $\theta = 0, \pi, 2\pi$ (iv) $6\pi a^2$ units²

N2019 P1 Q10

N2019 P1 Q10

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