Abril Huynh

2023-03-26

Find an equation equivalent to ${x}^{2}-{y}^{2}=4$ in polar coordinates.

unnlattpdui

How to determine the polar coordinates:
We have,
${x}^{2}-{y}^{2}=4$
The polar coordinates can be discovered using,
Substitute $x=r\mathrm{cos}\left(\theta \right)$ and $y=r\mathrm{sin}\left(\theta \right)$
${\left(r\mathrm{cos}\left(\theta \right)\right)}^{2}-{\left(r\mathrm{sin}\left(\theta \right)\right)}^{2}=4⇒{r}^{2}{\mathrm{cos}}^{2}\left(\theta \right)-{r}^{2}{\mathrm{sin}}^{2}\left(\theta \right)=4⇒{r}^{2}\left({\mathrm{cos}}^{2}\left(\theta \right)-{\mathrm{sin}}^{2}\left(\theta \right)\right)=4⇒{r}^{2}\left({\mathrm{cos}}^{2}\left(\theta \right)-\left(1-{\mathrm{cos}}^{2}\left(\theta \right)\right)\right)=4\left(\because {\mathrm{sin}}^{2}\left(\theta \right)=1-{\mathrm{cos}}^{2}\left(\theta \right)\right)⇒{r}^{2}\left({\mathrm{cos}}^{2}\left(\theta \right)-1+{\mathrm{cos}}^{2}\left(\theta \right)\right)=4⇒{r}^{2}\left(2{\mathrm{cos}}^{2}\left(\theta \right)-1\right)=4⇒{r}^{2}=\frac{4}{2{\mathrm{cos}}^{2}\left(\theta \right)-1}⇒r=\frac{2}{2}$
substitute $r=\frac{2}{2}$
$x=\frac{2\mathrm{cos}\left(\theta \right)}{2}$
$y=\frac{2\mathrm{sin}\left(\theta \right)}{2}$
Therefore, the polar coordinates are $\left(\frac{2\mathrm{cos}\left(\theta \right)}{2},\frac{2\mathrm{sin}\left(\theta \right)}{2}\right)$.

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