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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




Question Number 164520 by ZiYangLee last updated on 18/Jan/22
A circle is drawn with center (0,1) and radius 1.  A line OAB is drawn, making an angle θ 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  A and B  parallel to the x- and y-axes respectively  to intersect at P.   Prove that  (i) OA=2 sin θ     and   (ii)the coordinates of P are (2 cot θ, 2 sin^2 θ)  Hence, find the Cartesian equation of the locus of P.
$$\mathrm{A}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{with}\:\mathrm{center}\:\left(\mathrm{0},\mathrm{1}\right)\:\mathrm{and}\:\mathrm{radius}\:\mathrm{1}. \\ $$$$\mathrm{A}\:\mathrm{line}\:\mathrm{OAB}\:\mathrm{is}\:\mathrm{drawn},\:\mathrm{making}\:\mathrm{an}\:\mathrm{angle}\:\theta\:\mathrm{with}\:\mathrm{the} \\ $$$${x}-\mathrm{axis}\:\mathrm{to}\:\mathrm{cut}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{at}\:{A}\:\mathrm{and}\:\mathrm{the}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{circle}\:\mathrm{at}\:\left(\mathrm{0},\mathrm{2}\right)\:\mathrm{at}\:{B}.\:\mathrm{Lines}\:\mathrm{are}\:\mathrm{now}\:\mathrm{drawn}\:\mathrm{through} \\ $$$${A}\:\mathrm{and}\:{B}\:\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:{x}-\:\mathrm{and}\:{y}-\mathrm{axes}\:\mathrm{respectively} \\ $$$$\mathrm{to}\:\mathrm{intersect}\:\mathrm{at}\:{P}.\:\:\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{OA}=\mathrm{2}\:\mathrm{sin}\:\theta\:\:\:\:\:\mathrm{and}\: \\ $$$$\left(\mathrm{ii}\right)\mathrm{the}\:\mathrm{coordinates}\:\mathrm{of}\:{P}\:\mathrm{are}\:\left(\mathrm{2}\:\mathrm{cot}\:\theta,\:\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} \theta\right) \\ $$$$\mathrm{Hence},\:\mathrm{find}\:\mathrm{the}\:\mathrm{Cartesian}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{P}. \\ $$

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