Problem 52 - Algebra and Trigonometry

Question

Write a polar equation of the conic that is named and described. Hyperbola: a focus at the pole; directrix: \(x=-1 ; e=\frac{3}{2}\)

Step-by-Step Solution

Answer

The polar equation of the given hyperbola is \(r = \frac{\frac{3}{2}*1}{1+\frac{3}{2}\cos(\theta)}\).

Step 1: Identify the Given Properties

The given properties of the hyperbola are: focus at the pole (origin), directrix at x=-1, and eccentricity e=3/2.

Step 2: Determine Perpendicular Distance from the Pole to the Directrix

The pole is at the origin (0,0), and the directrix is a vertical line at x=-1. The perpendicular distance \(d\) between a point and a line in a coordinate geometry is simply the absolute difference along the axis. So \(d = |-1-0| = 1\).

Step 3: Formulate Polar Equation of a Hyperbola

Now plug the given eccentricity \(e=3/2\) and computed distance \(d=1\) into the standard form of the polar equation of a conic section \(r = \frac{ed}{1+e\cos(\theta)}\). The polar equation becomes \(r = \frac{\frac{3}{2}*1}{1+\frac{3}{2}\cos(\theta)}\).