By definition, the directrix is a line such that the distance from the focus to a point on the curve is the eccentricity times the distance from the line to the same point. We've put the equation in a form that expresses exactly that. The distance from the focus, -r, is 1/2 times the distance from the line x = -2. (r is always negative in this equation, so the distance from the focus is the positive number -r.) Therefore, the eccentricity is 1/2 and the directrix is x = -2.
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r = (-2)/(2 + cos θ)
r(2 + cos θ) = -2
2r + rcos θ = -2
2r + x = -2
2r = -2 - x
-r = (1/2)(x + 2)
By definition, the directrix is a line such that the distance from the focus to a point on the curve is the eccentricity times the distance from the line to the same point. We've put the equation in a form that expresses exactly that. The distance from the focus, -r, is 1/2 times the distance from the line x = -2. (r is always negative in this equation, so the distance from the focus is the positive number -r.) Therefore, the eccentricity is 1/2 and the directrix is x = -2.