Question

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

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

2020-12-15

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

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 $$r = \frac{ke}{1 - e cos \theta}$$

Step 6 For example: The polar equation for a conic with eccentricity $$(e = 1) \text{and directrix}\ x = -3$$ is $$r = \frac{3\times1}{1 -1 \times cos \theta}$$ $$r = \frac{3}{1 - cos \theta}$$

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 $$r = \frac{ke}{1 + e sin \theta}$$

Step 8 For example: The polar equation for a conic with eccentricity $$(e = 0.5) \text{and directrix}\ y = 2$$ is $$r = \frac{2\times0.5}{1 + 0.5 sin \theta}$$ $$r = \frac{1}{1 + 0.5 sin \theta}$$

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 $$r = \frac{ke}{1 - e sin \theta}$$

Step 10 For example: The polar equation for a conic with eccentricity $$(e = 0.5) \text{and directrix}\ y = - 4$$ is $$r = \frac{4\times0.5}{1 - 0.5 sin \theta}$$ $$r = \frac{2}{1 - 0.5 sin \theta}$$