The polar equation r = 11/(2-9sinx) represents an ellipse.

To determine the location of the directrix from the focus located at the pole, we need to convert the equation to rectangular form. Using the conversion formulas, we can rewrite the equation as:

x = rcos(x) = 11cos(x)/(2 - 9sin(x))
y = rsin(x) = 11sin(x)/(2 - 9sin(x))

Simplifying these equations, we get:

(2 - 9sin(x))x - 11cos(x) = 0
(2 - 9sin(x))y - 11sin(x) = 0

Now, compare this equation with the standard equation of an ellipse:

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

We can see that A = 2 - 9sin(x), B = 0, C = 2 - 9sin(x), D = -11cos(x), E = -11sin(x), and F = 0.

Since B^2 - 4AC = 0 - 4(2 - 9sin(x))(2 - 9sin(x)) = -4(4 - 36sin(x) + 81sin^2(x)) < 0 for any value of x, the conic section is an ellipse.

As for the location of the directrix, for an ellipse it is not a straight line like for the standard equations of other conic sections (such as parabola or hyperbola). The focus and directrix for an ellipse are not directly related like in those cases. Instead, the directrices of an ellipse are located at equal distances on either side of the center, along the major axis. The distance between each directrix and the center is given by the equation:

d = a^2 / c

where a is the semi-major axis and c is the distance between the center and the focus.

Since the polar equation does not give us enough information about the values of a and c, we cannot determine the exact location of the directrix. To find the location of the directrix, we would need additional information.

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