Questions: Identify the conic of r=2/(2-cos θ)

Transcript text: Identify the conic of $r=\frac{2}{2-\cos \theta}$

Solution

Solution Steps

Step 1: Rewrite the equation in standard form

The given polar equation is $ r = \frac{2}{2 - \cos \theta} $. To identify the conic, we rewrite it in the standard form of a conic in polar coordinates: \[ r = \frac{ed}{1 - e \cos \theta}, \] where $ e $ is the eccentricity and $ d $ is the distance from the pole to the directrix.

Step 2: Compare with the standard form

Divide the numerator and denominator of the given equation by 2 to match the standard form: \[ r = \frac{1}{1 - \frac{1}{2} \cos \theta}. \] From this, we can see that: \[ e = \frac{1}{2}, \quad ed = 1. \]

Step 3: Determine the type of conic

Since $ e = \frac{1}{2} < 1 $, the conic is an ellipse. The value of $ e $ being less than 1 indicates that the conic is closed and bounded.

Step 4: Identify the conic

The conic represented by the equation $ r = \frac{2}{2 - \cos \theta} $ is an ellipse.