Problem 48 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Given the focus is at the pole, I can write the polar equation of a conic section if I know its eccentricity and the rectangular equation of the directrix.

Step-by-Step Solution

Answer

The statement makes sense. Given the focus is at the pole, it is indeed possible to write the polar equation of a conic section with the knowledge of its eccentricity and the rectangular equation of the directrix. These parameters uniquely define a conic section.

Step 1: Understand terminology

The focus of a conic section refers to a point, at which, for example, rays reflected off a parabolic surface converge. The directrix of a conic section is a line that defines the geometry of the section together with the focus. The eccentricity is a measure of how much the conic section deviates from being a circle.

Step 2: Evaluate the statement

The statement makes sense because if the focus is at the pole, knowing the eccentricity and the equation of the directrix can enable the determination of the polar equation of the conic section. This is due to the definition of eccentricity, which provides a relationship between the distance of a point on the section from the focus and from the directrix. Knowing this relationship and the equation of the directrix therefore, would define the conic section uniquely. The rectangular equation of the directrix would provide the additional necessary information about the position and orientation of the directrix.

Step 3: Formulate the conclusion

So, as long that focus is at the pole, the statement about being able to write the polar equation of a conic section if the eccentricity and the rectangular equation of the directrix are known is correct. However, the statement doesn't implicate whether this process is easy or hard, as writing the equation might be a complex process depending on the eccentricity and the equation of the directrix.