Problem 36 - Algebra and Trigonometry

Question

If you are given the standard form of the polar equation of a conic, how do you determine the location of a directrix from the focus at the pole?

Step-by-Step Solution

Answer

To find the location of the directrix from the focus at the pole in a conic's standard polar equation, one must identify the eccentricity \(e\) and the distance from the focus to the directrix \(p\), as well as set the appropriate angle \( \theta \). The location can then be calculated with the equation \( \dfrac{ep}{1+\cos(\theta)} \).

Step 1: Identifying the eccentricity and distance

Identify the values of the eccentricity \(e\) and the distance from the focus to the directrix \(p\). These values are given by the standard form of the polar equation of the given conic.

Step 2: Setting the appropriate angle

Set the value of the angle \( \theta \). In general, to obtain the equation of the directrix, we don't need a specific value for the angle. However, if the location of a specific directrix is requested, then the appropriate angle must be used. This angle is the one formed by the x-axis and the line connecting the focus and any point on the conic.

Step 3: Use the equation

Use the equation \( \dfrac{ep}{1+\cos(\theta)} \) to find the location of the directrix. This equation allows us to find the specific location of the directrix relative to the focus that is at the pole.