What are the standard equations for lines and conic sections in polar coordinates? Give examples.

Step 1 The standard polar equation for lines is given by $rcos(\theta -{\theta }_{\circ })=r_{\circ }$ where $(r_{\circ },{\theta }_{\circ })\text{is the foot of the | from the (0, 0) to the line and}{r}_{\circ }\ge 0$

Step 2 For example: If $(4,{45}^{\circ })$ is the foot of the perpendicular, then the polar equation of the line is $r\cos (\theta -{45}^{\circ })=4$

Step 3 The standard polar equation for a conic with eccentricity (e) and directrix $x=k$ and focus at the (0, 0) is given by $r=\frac{ke}{1+e\cos \theta }$

Step 4 For example: The polar equation for a conic with eccentricity $(e=1)\text{and directrix}x=5$ is $r=\frac{5\times 1}{1+1\times cos\theta }$ $r=\frac{5\times 1}{1+\cos \theta }$

Step 5 The standard polar equation for a conic with eccentricity (e) and directrix $x=-k$ and focus at the (0, 0) is given by $r=\frac{ke}{1-ecos\theta }$

Step 6 For example: The polar equation for a conic with eccentricity $(e=1)\text{and directrix}x=-3$ is $r=\frac{3\times 1}{1-1\times cos\theta }$ $r=\frac{3}{1-cos\theta }$

Step 7 The standard polar equation for a conic with eccentricity (e) and directrix $y=k$ and focus at the (0, 0) is given by $r=\frac{ke}{1+esin\theta }$

Step 8 For example: The polar equation for a conic with eccentricity $(e=0.5)\text{and directrix}y=2$ is $r=\frac{2\times 0.5}{1+0.5sin\theta }$ $r=\frac{1}{1+0.5sin\theta }$

Step 9 The standard polar equation for a conic with eccentricity (e) and directrix $y=-k$ and focus at the (0, 0) is given by $r=\frac{ke}{1-esin\theta }$

Step 10 For example: The polar equation for a conic with eccentricity $(e=0.5)\text{and directrix}y=-4$ is $r=\frac{4\times 0.5}{1-0.5sin\theta }$ $r=\frac{2}{1-0.5sin\theta }$