A-circle-is-drawn-with-center-0-1-and-radius-1-A-line-OAB-is-drawn-making-an-angle-with-the-x-axis-to-cut-the-circle-at-A-and-the-tangent-to-the-circle-at-0-2-at-B-Lines-are-now-drawn-through

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Question Number 164520 by ZiYangLee last updated on 18/Jan/22

$$\mathrm{A}\mathrm{circle}\mathrm{is}\mathrm{drawn}\mathrm{with}\mathrm{center}(0,1)\mathrm{and}\mathrm{radius}1.$$$$\mathrm{A}\mathrm{line}\mathrm{OAB}\mathrm{is}\mathrm{drawn},\mathrm{making}\mathrm{an}\mathrm{angle}\theta \mathrm{with}\mathrm{the}$$$$x-\mathrm{axis}\mathrm{to}\mathrm{cut}\mathrm{the}\mathrm{circle}\mathrm{at}A\mathrm{and}\mathrm{the}\mathrm{tangent}\mathrm{to}\mathrm{the}$$$$\mathrm{circle}\mathrm{at}(0,2)\mathrm{at}B.\mathrm{Lines}\mathrm{are}\mathrm{now}\mathrm{drawn}\mathrm{through}$$$$A\mathrm{and}B\mathrm{parallel}\mathrm{to}\mathrm{the}x-\mathrm{and}y-\mathrm{axes}\mathrm{respectively}$$$$\mathrm{to}\mathrm{intersect}\mathrm{at}P.\mathrm{Prove}\mathrm{that}$$$$\left(\mathrm{i}\right)\mathrm{OA}=2\sin \theta \mathrm{and}$$$$\left(\mathrm{ii}\right)\mathrm{the}\mathrm{coordinates}\mathrm{of}P\mathrm{are}(2\cot \theta ,2{\sin }^2\theta )$$$$\mathrm{Hence},\mathrm{find}\mathrm{the}\mathrm{Cartesian}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{locus}\mathrm{of}P.$$

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