# Provide the standard equations for lines and conic sections in

Provide the standard equations for lines and conic sections in polar coordinates with examples.
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Gennenzip
The standard equation for lines in polar coordinates.
If the is the foot of the perpendicular from the origin to the line L, and ${r}_{0}\ge 0$, then an equation for L is $\mathrm{cos}\left(\theta -{\theta }_{0}\right)={r}_{0}$.
Example:
Consider ${\theta }_{0}=\frac{\pi }{3}$ and ${r}_{0}=2.$
The equation for line is $r\mathrm{cos}\left(\theta -{\theta }_{0}\right)={r}_{0}$
Substitute $\frac{\pi }{3}$ for ${\theta }_{0}$ and 2 for ${r}_{0}$
$r\mathrm{cos}\left(\theta -\frac{\pi }{3}\right)=2$
$r\left(\mathrm{cos}\theta \mathrm{cos}\frac{\pi }{3}+\mathrm{sin}\theta \mathrm{sin}\frac{\pi }{3}\right)=2$
$\frac{1}{2}r\mathrm{cos}\theta +\frac{\sqrt{3}}{2}r\mathrm{sin}\theta =2$
$r\mathrm{cos}\theta +\sqrt{3}r\mathrm{sin}\theta =4$
Substitute x for $r\mathrm{cos}\theta$ and y for $r\mathrm{sin}\theta$
$x+\sqrt{3y}=4$
Thus, the equation of line is $x+\sqrt{3y}=4$
The equation for a conic with eccentricity e is
$r=\frac{ke}{1+e\mathrm{cos}\theta }$
where, $x=k>0$ is the vertical directrix.
Example:
Consider the equation of hyperbola with the eccentricity $\frac{3}{2}$ and directrix $x=2$
The equation for a conic with eccentricity e is $r=\frac{ke}{1+e\mathrm{cos}\theta }$
Substitute 2 for k and $\frac{3}{2}$ for e.
$r=\frac{2×\frac{3}{2}}{1+\frac{3}{2}\mathrm{cos}\theta }$
$=\frac{3}{\frac{2+3\mathrm{cos}\theta }{2}}$
$=\frac{6}{2+3\mathrm{cos}\theta }$
Hence, the equation for a conic is $r=\frac{6}{2+3\mathrm{cos}\theta }$