Problem 48 - College Algebra

Question

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-1 ; e=1\)

Step-by-Step Solution

Answer

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

Step 1: Identify Conic Type

The eccentricity is given as \( e = 1 \). An eccentricity of 1 indicates that the conic is a parabola.

Step 2: Use the Polar Equation of a Parabola

The polar equation of a conic with focus at the origin and given directrix is \( r = \frac{ed}{1 + e\cos\theta} \) when the directrix is vertical. Since \( e = 1 \), substitute these into the formula.

Step 3: Express the Directrix

The directrix given is \( x = -1 \). For the equation, \( d = 1 \) as the directrix distance to the pole (origin) is 1 unit away in the negative x-direction. This implies we will substitute \( d = 1 \) into the formula.

Step 4: Substitute Values into Polar Equation

Substituting \( e = 1 \) and \( d = 1 \) into the equation \( r = \frac{ed}{1 + e\cos\theta} \) gives: \[ r = \frac{1 \cdot 1}{1 + 1 \cos\theta} = \frac{1}{1 + \cos\theta} \].

Step 5: Verify and Simplify the Result

Check that the equation is correct for a parabola with a directrix at \( x = -1 \). The derived equation \( r = \frac{1}{1 + \cos\theta} \) is indeed the representation of a parabola centered at the pole with the specified directrix.

Key Concepts

Eccentricity

Eccentricity is a fundamental concept in the study of conic sections. It determines the shape of the conic: the value of the eccentricity, denoted as \( e \), helps identify whether the conic is a circle, ellipse, parabola, or hyperbola. Here's what you need to know about different conic sections based on eccentricity:

• If \( e = 0 \), the conic is a circle.
• For \( 0 < e < 1 \), it is an ellipse.
• An eccentricity of \( e = 1 \) characterizes a parabola.
• If \( e > 1 \), the conic is a hyperbola.

In the given exercise, the eccentricity \( e = 1 \) indicates that the conic is a parabola.This means that the distance from any point on the parabola to the focus equals its distance to the directrix.

Parabola

A parabola is a unique type of conic section that has an eccentricity \( e = 1 \). It is defined as the set of all points equidistant from a point known as the focus, and a line called the directrix.A polar equation for a parabola can be derived when it's described in terms of a radial distance \( r \) and an angle \( \theta \), with its focus located at the origin. The general form of the polar equation is: \[ r = \frac{ed}{1 + e\cos\theta} \].For the exercise, we considered a parabola with directrix perpendicular to the polar axis.The radial distance reflects how the conic section changes as the angle \( \theta \) varies, showing symmetry typical of a parabola.

Directrix

The directrix is a crucial element when dealing with conic sections, as it helps define their shape.It is a fixed line that, together with the focus, determines the position and curvature of the conic.In a parabola, each point on the curve is equidistant from the focus and the directrix.

• For our exercise, the directrix is given as \( x = -1 \).
• This means it's a vertical line, 1 unit away from the origin in the negative x-direction.

The directrix interacts with the polar equation to create a formula: \( r = \frac{ed}{1 + e\cos\theta} \).Here, \( d \) represents the distance from the pole to the directrix.

Focus

In conic sections, the focus is a key point from which distances are measured to define the curve.For a parabola, the focus is not just any point but the point at which distances are equidistant to both it and the directrix.

• In the exercise, the focus is positioned at the origin in polar coordinates, \((0,0)\).
• This simplified the derivation of the polar equation of the conic.

The polar form of conics with a focus at the origin typically enhances understanding and application in real-world problems, where such symmetry and positioning are relevant.

Conic Sections

Conic sections arise from the intersection of a plane with a double-napped cone.This intersection can produce various shapes like circles, ellipses, parabolas, and hyperbolas, each being uniquely characterized by its eccentricity.The nature of conic sections lays the groundwork for broader mathematical fields and applications, emphasizing their wide relevance.

• The parabola, characterized by \( e = 1 \), plays a vital role in fields like physics, engineering, and nature.
• Ellipses and hyperbolas also have important real-world applications, including orbits of planets and hyperbolic geometry.

Understanding these shapes requires a foundational grasp of concepts like eccentricity, focus, and directrix.