Problem 41 - Essential Calculus

Question

Identify each conic. (a) \(r=\frac{5}{1-2 \cos \theta}\) (b) \(r=\frac{5}{10-\sin \theta}\) (c) \(r=\frac{5}{3-3 \cos \theta}\) (d) \(r=\frac{5}{1-3 \sin (\theta-\pi / 4)}\)

Step-by-Step Solution

Answer

(a) Hyperbola, (b) Ellipse, (c) Parabola, (d) Hyperbola.

Step 1: Identifying the conic for Eq. (a)

The standard form of a conic section in polar coordinates is \( r = \frac{e}{1- \pm e \cos (\theta - \phi)} \), where e is the eccentricity of the conic section. For equation (a), \(r= \frac{5}{1-2 \cos \theta} \), the eccentricity e = 2. Since e > 1, this conic section is a hyperbola.

Step 2: Identifying the conic for Eq. (b)

For equation (b), \(r= \frac{5}{10- \sin \theta} \), the standard form resembles \(r = \frac {e}{d - e \sin \theta} \). While not the exact standard form for a conic section, this equation can be simplified to normalize a term (like in standard form). However, it's notable that whatever the eccentricity realized from simplifying, it will be less than 1 as the denominator is larger than the numerator. Hence, this conic section is an ellipse.

Step 3: Identifying the conic for Eq. (c)

For equation (c), \(r= \frac{5}{3-3 \cos \theta} \), the eccentricity e = 1. Since e = 1, this conic section is a parabola.

Step 4: Identifying the conic for Eq. (d)

For equation (d), \(r= \frac{5}{1-3 \sin (\theta- \pi/4)} \), after re-writing and comparing to the standard form for a conic section, the eccentricity realized is 3. Since e > 1, this conic section is a hyperbola.