91Ó°ÊÓ

Polar coordinates offer an alternative to Cartesian coordinates for plotting points in a plane. They consist of a pair, \(r, \ heta\), where \r\ is the distance from the origin and \theta\ is the angle measured from the positive x-axis. This system is particularly useful for describing curves and shapes centered around a point or involving angles. Conic sections often have simpler equations in polar coordinates, which is why they're commonly used in these scenarios. In the exercise, we see a conic expressed in polar form as \(r = \rac{6}{1 - 2\cos\theta}\). This form allows us to immediately recognize the type of conic and important characteristics like eccentricity and directrix.

Eccentricity is a critical parameter in defining the shape of conic sections. It is denoted by \e\ and tells us how much a conic section deviates from being circular. The value of eccentricity determines the nature of the conic:

• If \(0 < e < 1\), the conic is an ellipse.
• If \(e = 1\), it is a parabola.
• If \(e > 1\), it is a hyperbola.

In our problem, with the equation \(r = \rac{6}{1 - 2\cos\theta}\), the eccentricity is found to be \e = 2\. Since \(e > 1\), the given conic is a hyperbola, which naturally follows from its definition as having two separate branches.

The directrix is a line related to a conic section that helps maintain its geometric properties with respect to a focus. Each type of conic section - ellipse, parabola, or hyperbola - uses the directrix in its definition. The relationship between the focus, directrix, and eccentricity is crucial to understanding conics.For a hyperbola described in polar coordinates by \(r = \rac{ed}{1 - e\cos\theta}\), the term \(ed\) represents the product of eccentricity and the distance from the pole to the directrix. From the given problem, this product is 6, given the equation is \(r = \rac{6}{1 - 2\cos\theta}\). With \e = 2\, solving \(2d = 6\) gives us \d = 3\. Therefore, the directrix is located at \x = -3\.

A hyperbola is a type of conic section that is defined as the locus of all points where the difference of distances from two fixed points, known as foci, is constant. Unlike an ellipse, a hyperbola forms two separate curves called branches. It can open horizontally or vertically in the plane. In polar coordinates, the general equation of a hyperbola is given as \(r = \rac{ed}{1 \pm e\cos\theta}\) or \(r = \rac{ed}{1 \pm e\sin\theta}\).In this exercise, the hyperbola has a polar equation \(r = \rac{6}{1 - 2\cos\theta}\), where the negative cosine indicates a horizontal directrix, which results in a horizontally opening hyperbola. Key features that characterize a hyperbola include its branches, asymptotes, foci, and of course, its eccentricity which, in this case, confirms the hyperbola with \(e = 2\).