Problem 35 - Algebra and Trigonometry

Question

If you are given the standard form of the polar equation of a conic, how do you determine its eccentricity?

Step-by-Step Solution

Answer

The eccentricity of the conic section can be determined by rearranging the polar form of the equation \(r=\frac{e p}{1+e cos(\theta)}\) or \(r=\frac{e p}{1+e sin(\theta)}\) to isolate and solve for the eccentricity, e.

Step 1: Identify the Polar Equation

Identify the given polar form of the equation of a conic section. It can be in the form of \(r=\frac{e p}{1+e cos(\theta)}\) or \(r=\frac{e p}{1+e sin(\theta)}\) depending on the conic section.

Step 2: Rearrange the Equation

Rearrange the equation to isolate e or eccentricity on one side. Multiply both sides of the equation by \(1+e cos(\theta)\) or \(1+e sin(\theta)\) to eliminate denominator.

Step 3: Calculate Eccentricity

Calculate the eccentricity, e, by dividing \(r(1+e cos(\theta))\) or \(r(1+e sin(\theta))\) by p. The value of e determines the type of the conic section (e = 0 for circle, 01 for hyperbola).