Identify the conic whose equation is given. ellipse parabola hyperbola

Hyperbola  

Step-by-step 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.

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