Provide the standard equations for lines and conic sections in

The standard equation for lines in polar coordinates.
If the ${\displaystyle P_0(r_0,{\theta }_0)}$ is the foot of the perpendicular from the origin to the line L, and ${\displaystyle r_0\ge 0}$, then an equation for L is ${\displaystyle \cos (\theta -{\theta }_0)=r_0}$.
Example:
Consider ${\displaystyle {\theta }_0=\frac{\pi }{3}}$ and ${\displaystyle r_0=2.}$
The equation for line is ${\displaystyle r\cos (\theta -{\theta }_0)=r_0}$
Substitute ${\displaystyle \frac{\pi }{3}}$ for ${\displaystyle {\theta }_0}$ and 2 for ${\displaystyle r_0}$
${\displaystyle r\cos (\theta -\frac{\pi }{3})=2}$
${\displaystyle r(\cos \theta \cos \frac{\pi }{3}+\sin \theta \sin \frac{\pi }{3})=2}$
${\displaystyle \frac{1}{2}r\cos \theta +\frac{\sqrt{3}}{2}r\sin \theta =2}$
${\displaystyle r\cos \theta +\sqrt{3}r\sin \theta =4}$
Substitute x for ${\displaystyle r\cos \theta }$ and y for ${\displaystyle r\sin \theta }$
${\displaystyle x+\sqrt{3y}=4}$
Thus, the equation of line is ${\displaystyle x+\sqrt{3y}=4}$
The equation for a conic with eccentricity e is
${\displaystyle r=\frac{ke}{1+e\cos \theta }}$
where, ${\displaystyle x=k>0}$ is the vertical directrix.
Example:
Consider the equation of hyperbola with the eccentricity ${\displaystyle \frac{3}{2}}$ and directrix ${\displaystyle x=2}$
The equation for a conic with eccentricity e is ${\displaystyle r=\frac{ke}{1+e\cos \theta }}$
Substitute 2 for k and ${\displaystyle \frac{3}{2}}$ for e.
${\displaystyle r=\frac{2\times \frac{3}{2}}{1+\frac{3}{2}\cos \theta }}$
${\displaystyle =\frac{3}{\frac{2+3\cos \theta }{2}}}$
${\displaystyle =\frac{6}{2+3\cos \theta }}$
Hence, the equation for a conic is ${\displaystyle r=\frac{6}{2+3\cos \theta }}$