Final answer:
The polar equation for the ellipse with focus at the origin, eccentricity 1/4, and directrix r = 2 secθ is r = ½ / (1 + 0.25·cosθ).
Step-by-step explanation:
To find a polar equation for the ellipse with a focus at the origin (0, 0), an eccentricity of 1/4, and a directrix at r = 2 secθ, we use the definition of the ellipse in polar coordinates. The general equation for a conic in polar coordinates, where e is the eccentricity, r is the radius, and θ is the angle, is given by:
r = rac{ed}{1 + e · cos θ}
where d is the distance to the directrix. Since the directrix is given by r = 2 secθ, we have d = 2. Substituting e = 1/4 and d = 2 into the equation:
r = rac{(1/4) · 2}{1 + (1/4) · cos θ} = rac{1/2}{1 + (1/4) · cos θ}
This is the polar equation for an ellipse with the given parameters.