Final answer:
The eccentricity for the conic section represented by the equation r = 1/(4 + sin(theta)) corresponds to a parabola. Since a parabola has an eccentricity equal to 1, and the given equation cannot be manipulated to describe an ellipse or a hyperbola. Without additional information, the exact equation for the directrix cannot be provided, and a sketch of the conic would require plotting the equation in polar coordinates.
Step-by-step explanation:
The given equation, r = 1/(4 + sin(theta)) is analogous to the standard equation of a conic r = e / (1 + e*cos(theta)) where e is the eccentricity. By comparing, we see that e*cos(theta) is replaced with sin(theta), meaning the conic has been rotated, but for finding the eccentricity, this does not affect the comparison.
To find the eccentricity e, we need to find a way to express our equation in a form that allows us to isolate e. The given equation cannot be an ellipse, because for an ellipse, the eccentricity must be between 0 and 1, and it does not represent a hyperbola since the eccentricity must be greater than 1 for a hyperbola. It also cannot be a circle as a circle's eccentricity is 0.
Since the given equation has no e*cos(theta) term and cannot be manipulated to form one, it indicates that the equation does not describe an ellipse or a hyperbola. Instead, it corresponds to a parabola since a parabola is defined by an eccentricity equal to 1. The equation of the directrix for a parabola in Cartesian coordinates is typically given by y = k or x = k where k is a constant. However, without additional information or transformation to Cartesian coordinates, finding an exact equation for the directrix is not possible from the given polar equation.
To sketch the conic, it would be necessary to plot the curve in a polar coordinate system, where each point on the curve is at a distance r from the pole (origin) at an angle theta. Nonetheless, without further manipulation to transform the equation into Cartesian coordinates, a detailed sketch is beyond this explanation.