Question

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

Conic sections
ANSWERED
asked 2020-12-14
What are the standard equations for lines and conic sections in polar coordinates? Give examples.

Answers (1)

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 \(r = \frac{4\times0.5}{1 - 0.5 sin \theta}\) \(r = \frac{2}{1 - 0.5 sin \theta}\)

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