Problem 51 - Algebra and Trigonometry

Question

Write a polar equation of the conic that is named and described. Ellipse: a focus at the pole; vertex: \((4,0) ; e=\frac{1}{2}\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{4}{1 - \frac {cos(\theta)}{2} } \)

Step 1: Identify Parameters

The eccentricity, e, is given directly as \( e = \frac{1}{2} \). The vertex is the point (4,0), so the distance from the focus to the vertex (the semi-major axis), denoted by a, is 4.

Step 2: Compute the distance from the focus to the directrix, p

In an ellipse with a focus at the origin, the relationship between a, e, and p is given by \( a = \frac {ep}{1-e} \). We know a and e, so we can solve for p: \( a = \frac {ep}{1-e} \) becomes \( 4 = \frac {(\frac{1}{2})p}{1 - \frac{1}{2}} \) which simplifies to \( 4 = \frac {p}{2} \) or \( p = 8 \)

Step 3: Write the Polar Equation

Substitute the values of e and p into the general equation \( r = \frac{ep}{1 - ecos(\theta)} \) this gives \( r = \frac{( \frac{1}{2} * 8 )}{ 1 - ( \frac{1}{2} * cos(\theta) ) } \), simplifying to \( r = \frac{4}{1 - \frac {cos(\theta)}{2} } \)