Problem 43 - Precalculus with Limits

Question

In Exercises 39-54, find a polar equation of the conic with its focus at the pole. \(\textit{Conic}\) Hyperbola \(\textit{Eccentricity}\) \(e=2\) \(\textit{Directrix}\) \(x=1\)

Step-by-Step Solution

Answer

The polar coordinate equation of a hyperbola is \(r = \frac{2}{1 \pm 2 \cos(\theta)}\).

Step 1: Identify the distance from directrix to the pole

The directrix \(x = 1\) implies that the distance \(d\) from the directrix to the pole (or origin in Cartesian system) is 1.

Step 2: Identify the eccentricity

From the problem statement, the eccentricity \(e\) of the hyperbola is given as 2.

Step 3: Create the equation of conic in polar coordinates

Now, substitute \(d = 1\) and \(e = 2\) into the formula \(r = \frac{ed}{1 ± e \cos(\theta)}\) OR \(r = \frac{ed}{1 ± e \sin(\theta)}\) depending upon the axis of the hyperbola. Here, since directrix \(x = 1\) is vertical line, we choose \(r = \frac{ed}{1 ± e \cos(\theta)}\). Therefore, the polar equation of the hyperbola is \(r = \frac{2 * 1}{1 \pm 2 \cos(\theta)} = \frac{2}{1 \pm 2 \cos(\theta)}\).

Key Concepts

Hyperbola in Polar Coordinates

Understanding a hyperbola in polar coordinates can initially seem daunting, but let's break it down to its basic principles. A hyperbola is a type of conic section, defined as the set of points where the difference between the distances to two fixed points (known as foci) is constant. When we describe a hyperbola in polar coordinates, we focus on one of these points (the focus) being at the pole, which is the origin of our polar system.

For a hyperbola centered at the pole, the polar equation is typically written as \( r = \frac{ed}{1 \pm e \cos(\theta)} \) or \( r = \frac{ed}{1 \pm e \sin(\theta)} \) depending on its orientation. In our exercise, the hyperbola has a vertical directrix, so we use the cosine form. Here, \( e \) represents the eccentricity while \( d \) is the perpendicular distance from the directrix to the pole. By plugging \( e = 2 \) and \( d = 1 \) into the former equation, we obtain \( r = \frac{2}{1 \pm 2 \cos(\theta)} \) which describes a hyperbola with a focus at the origin in polar coordinates.

This understanding allows us to visualize the hyperbola's shape through its reflection property with respect to the pole and directrix.

Eccentricity of Conic Sections

Eccentricity is a measure describing the deviation of a conic section from being circular. For any conic section, the eccentricity \( e \) is defined as the ratio of the distance from any point on the conic to a focus, to the perpendicular distance from that point to the nearest directrix.

In general, the eccentricity can be understood as:

• \(e = 0\) for a circle, which can be thought of as a special case of an ellipse with an eccentricity of zero.
• \(0 < e < 1\) for an ellipse, which has an oval shape.
• \(e = 1\) for a parabola, which is shaped like a U or V.
• \(e > 1\) for a hyperbola, indicating it is more 'stretched' away from being circular.

The higher the eccentricity, the less the conic section resembles a circle. Our exercise features a hyperbola with an eccentricity of 2, which confirms that the shape is quite elongated since it's greater than 1.

Directrix of Conic Sections

The directrix of a conic section is a line which, together with the focus, helps to define the conic. It is crucial for calculating distances used in the definitions of ellipses, parabolas, and hyperbolas.

Each conic section has at least one directrix, and it serves as a reference line from which distances are measured to points on the curve. For instance:

• In a parabola, each point is equidistant from the focus and the directrix.
• In an ellipse and a hyperbola, the ratio of the distance from a point on the curve to the focus, and to the directrix, is constant and equal to the eccentricity.

In our specific example, the directrix is given by the equation \( x = 1 \), which means it's a vertical line one unit away from the pole (origin). This line is fundamental when deriving the polar equation of our hyperbola since the distance between the directrix and the pole, denoted as \( d \) in the equation \( r = \frac{ed}{1 \pm e \cos(\theta)} \), is directly used in its formulation.