# Find all points of interception of the pair of polar equations in polar form r=3 and r=-4cosx

Question
Equations
Find all points of interception of the pair of polar equations in polar form r=3 and $$\displaystyle{r}=-{4}{\cos{{x}}}$$

2020-12-16
Step 1
Given polar equations:
r=3
$$\displaystyle{r}=-{4}{\cos{{\left({0}\right)}}}$$
Step 2
To find points of intersection of given polar equations:
$$\displaystyle-{4}{\cos{{\left({0}\right)}}}={3}$$
$$\displaystyle{\cos{{\left({0}\right)}}}=-\frac{{3}}{{4}}$$
$$\displaystyle{0}={{\cos}^{{-{1}}}{\left(-\frac{{3}}{{4}}\right)}}$$
$$\displaystyle{0}={138.6}^{{\circ}},{221.4}^{{\circ}}$$
Step 3
Therefore,
The points of intersection of the pair of polar equations are
$$\displaystyle{\left({3},{138.6}^{{\circ}}\right)}$$
$$\displaystyle{\left({3},{221.4}^{{\circ}}\right)}$$

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$$\displaystyle{x}'_{{1}}{\left({t}\right)}={3}{x}_{{1}}{\left({t}\right)}-{2}{x}_{{2}}{\left({t}\right)}+{e}^{{t}}{x}_{{3}}{\left({t}\right)}$$
$$\displaystyle{x}'_{{2}}{\left({t}\right)}={\sin{{\left({t}\right)}}}{x}_{{1}}{\left({t}\right)}+{\cos{{\left({t}\right)}}}{x}_{{3}}{\left({t}\right)}$$
$$\displaystyle{x}'_{{3}}{\left({t}\right)}={t}^{{2}}{x}_{{1}}{\left({t}\right)}+{t}{x}^{{2}}{\left({t}\right)}+{x}_{{3}}{\left({t}\right)}$$