Question

Identify the conic whose equation is given.


`r= (4)/(2-4cos\theta )`

ellipse
parabola
hyperbola

Answer 1

Hyperbola

Explanation:

The polar equation of a conic section with directrix ± d has the standard form:

r=ed/(1 ± ecosθ)

where e = the eccentricity.

The eccentricity determines the type of conic section:

e = 0 ⇒ circle

0 < e < 1 ⇒ ellipse

e = 1 ⇒ parabola

e > 1 ⇒ hyperbola

Step 1. Convert the equation to standard form

r = 4/(2 – 4 cosθ)

Divide numerator and denominator by 2

r = 2/(1 - 2cosθ)

Step 2. Identify the conic

e = 2, so the conic is a hyperbola.

The polar plot of the function (below) confirms that the conic is a hyperbola.