Question

Find a polar equation for each conic section. Assume one focus is at the origin.

Short answer

Answer: The general polar equations for each conic section based on eccentricity are as follows: Ellipse (e < 1): \(r = \frac{a(1-e^2)}{1 + e \cos(\theta)}\) Parabola (e = 1): \(r = \frac{a}{1 + \cos(\theta)}\) Hyperbola (e > 1): \(r = \frac{a(e^2 - 1)}{1 - e \cos(\theta)}\)

Step-by-step solution

Analyze each conic section and its standard Cartesian equations

An ellipse has the standard form: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ A parabola has the standard form: $$y = ax^2+bx+c$$ A hyperbola has the standard form: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

Write the polar coordinate conversions

To convert Cartesian to polar coordinates, we use the following conversions: $$x = r \cos(\theta)$$ $$y = r \sin(\theta)$$

Write the conic sections' equations in polar form

Ellipse: Replace \(x\) and \(y\) with their polar conversions in the standard form of an ellipse: $$\frac{(r \cos(\theta))^2}{a^2} + \frac{(r \sin(\theta))^2}{b^2} = 1$$ Simplify the equation to find the general polar equation: $$r^2 = \frac{a^2b^2}{a^2 \sin^2(\theta) + b^2 \cos^2(\theta)}$$ Parabola: Replace \(x\) and \(y\) with their polar conversions in the standard form of a parabola: $$ r\sin(\theta) = a(r \cos(\theta))^2 + b(r \cos(\theta)) + c$$ Simplify the equation to find the general polar equation: $$ r = \frac{c}{\sin(\theta) - a \cos^2(\theta) - b \cos(\theta)}$$ Hyperbola: Replace \(x\) and \(y\) with their polar conversions in the standard form of a hyperbola: $$\frac{(r \cos(\theta))^2}{a^2} - \frac{(r \sin(\theta))^2}{b^2} = 1$$ Simplify the equation to find the general polar equation: $$ r^2 = \frac{a^2b^2}{a^2 \sin^2(\theta) - b^2 \cos^2(\theta)}$$

Express the polar equations in terms of eccentricity

Conic sections can also be described using eccentricity (e). Here are the general polar equations for each conic section based on eccentricity: Ellipse (e < 1): $$r = \frac{a(1-e^2)}{1 + e \cos(\theta)}$$ Parabola (e = 1): $$r = \frac{a}{1 + \cos(\theta)}$$ Hyperbola (e > 1): $$r = \frac{a(e^2 - 1)}{1 - e \cos(\theta)}$$ These are the general polar equations for each conic section considering one focus at the origin.