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etude-complete-de-la-courbe-d-equation-polaire-r-1-sin-2-symetrie-et-trace-




Question Number 144900 by ArielVyny last updated on 30/Jun/21
etude complete de la courbe d′equation  polaire r=(1/(sin(2θ)))    (symetrie et trace)
$${etude}\:{complete}\:{de}\:{la}\:{courbe}\:{d}'{equation} \\ $$$${polaire}\:{r}=\frac{\mathrm{1}}{{sin}\left(\mathrm{2}\theta\right)}\:\:\:\:\left({symetrie}\:{et}\:{trace}\right) \\ $$$$ \\ $$
Answered by qaz last updated on 30/Jun/21
2rsin θcos θ=1  ⇒2r^2 sin θcos θ=r  ⇒2xy=(√(x^2 +y^2 ))  ⇒x^2 +y^2 −4x^2 y^2 =0
$$\mathrm{2rsin}\:\theta\mathrm{cos}\:\theta=\mathrm{1} \\ $$$$\Rightarrow\mathrm{2r}^{\mathrm{2}} \mathrm{sin}\:\theta\mathrm{cos}\:\theta=\mathrm{r} \\ $$$$\Rightarrow\mathrm{2xy}=\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} } \\ $$$$\Rightarrow\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} −\mathrm{4x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} =\mathrm{0} \\ $$

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