Problem 32 - Thomas Calculus

Question

Exercises \(29-36\) give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$e=2, \quad x=4$$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{8}{1 + 2 \cos\theta} \).

Step 1: Identify Eccentricity and Directrix

The eccentricity is given as \( e = 2 \). This indicates that the conic is a hyperbola, as eccentricities greater than 1 correspond to hyperbolas. The directrix is given by \( x = 4 \).

Step 2: Understand the Polar Equation Form

For conic sections with a focus at the origin, the general polar form is \( r = \frac{ed}{1 \pm e \cos\theta} \) for horizontal directrices. We use \(+\) for conics with horizontal directrices right of the pole since \( x = 4 \) is positive.

Step 3: Determine the Direction and Parameters

Since the directrix is horizontal (\( x = 4 \)), the form \( r = \frac{ed}{1 + e \cos\theta} \) is appropriate, where \( d \) is the distance from the origin to the directrix.

Step 4: Substitute Known Values

Given \( e = 2 \) and \( d = 4 \), substitute these into the polar equation: \[ r = \frac{2 \times 4}{1 + 2 \cos\theta} \] Simplifying gives the polar equation for the hyperbola.

Key Concepts

Conic Sections

Conic sections are a family of curves obtained by intersecting a plane with a cone. There are four main types of conic sections:

• Circle: Every point on the curve is equidistant from a center point. Formed when the plane cuts parallel to the base of the cone.
• Ellipse: An elongated circle with two foci. Occurs when the plane cuts through the cone at an angle but doesn't go through the base.
• Parabola: Curved like an arch and formed as the plane cuts parallel to one side of the conical surface.
• Hyperbola: Consists of two separate arcs. It appears when the plane intersects both nappes of the cone.

Conic sections have unique properties and equations. When defined in polar coordinates, they exhibit forms that can describe their behavior relative to a focus and a directrix. These equations often incorporate the eccentricity to determine the type of conic section.

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. This system is especially useful for dealing with problems involving angles and distances from a central point.

• Reference Point: Known as the pole, and analogous to the origin in Cartesian coordinates.
• Angle: Usually measured in radians or degrees from a fixed direction, often the positive x-axis.
• Distance: Referred to as the radius, denoted by \( r \), which is the distance from the pole to the point.

Using polar coordinates simplifies the representation of conic sections, particularly when one focus is at the origin. The general polar equation for conic sections enables easier calculations and visualizations of curves such as ellipses and hyperbolas.

Eccentricity

Eccentricity is a numerical parameter that defines the shape of a conic section. It is denoted by \( e \) and has specific meanings for each conic:

• Circle: \( e = 0 \). Perfectly circular shape.
• Ellipse: \( 0 < e < 1 \). Oval shape with two foci.
• Parabola: \( e = 1 \). Curved in nature and characterized by having a single focus and a directrix.
• Hyperbola: \( e > 1 \). Divided into two mirrored arcs, showing a more stretched shape.

In polar coordinates, eccentricity helps to form the equation of the conic section. For the given exercise, the eccentricity \( e = 2 \) indicates a hyperbola. This value instrumental in defining features such as the size and the overall shape of the conic.