Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Parabola} &e=1&x=-1\end{array}$$

Step-by-Step Solution

Answer

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

Step 1: Recall the polar form of a conic equation

The general polar form of a conic section with its focus at the pole and directrix \(x = -D\) is \(r = \frac{D}{1+e \cos \theta}\), where \(e\) is the eccentricity. For a parabola, the eccentricity \(e\) is always 1.

Step 2: Substitute the given values

You are given that the directrix of the parabola is \(x = -1\), so the value of D is 1 (as D is always taken as a positive number, the equation of directrix will just provide the magnitude of D). Also, we know that for a parabola, eccentricity \(e\) is 1. Substituting \(e = 1\) and \(D = 1\) in our general polar equation gives \(r = \frac{1}{1 + \cos \theta}\).

Key Concepts

Polar Coordinate System

The polar coordinate system is an alternative to the more common Cartesian coordinate system. Instead of using x and y coordinates to define a point's position, the polar system uses the distance from a reference point (called the pole, similar to the origin in Cartesian coordinates) and an angle from a reference direction. A point's location is given as \( (r, \theta) \) where \( r \) is the radius or the distance from the pole, and \( \theta \) is the angle, usually measured in degrees or radians, from the positive x-axis.\
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To translate between polar and Cartesian coordinates, you can use the following equations: \( x = r \cos\theta \) for the x-coordinate and \( y = r \sin\theta \) for the y-coordinate. This system is particularly useful when dealing with problems that have circular or spiral symmetry, where the equations can be much simpler in polar form. Learning to visualize and draw points using polar coordinates is a vital skill for students to master when exploring advanced mathematical concepts, including conic sections.

Eccentricity of a Conic

Eccentricity, denoted as \( e \), is a parameter that describes how much a conic section deviates from being circular. A conic section refers to curves obtained as intersections of a plane with a cone, such as ellipses, parabolas, and hyperbolas.\
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For a circle, the eccentricity is 0 because it is perfectly symmetrical in all directions. An ellipse, which is an elongated circle, has an eccentricity greater than 0 but less than 1. The larger the eccentricity, the more elongated the ellipse. A parabola, which is the shape of our given problem, has an eccentricity equal to 1. This value signifies that the parabola's arms will continue to diverge as they extend from the focus. For a hyperbola, the eccentricity is greater than 1, indicating that the curve is even more open than that of a parabola.\
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The concept of eccentricity helps us to classify conic sections and to understand their properties better. It also plays a crucial role in the equations of conic sections in polar coordinates, as seen in the given problem.

Conic Sections in Polar Coordinates

When you're working with conic sections in the polar coordinate system, the equation of the conic can be directly related to its eccentricity and its orientation with respect to the pole. Parabolas have a very distinctive feature in that they only have one focus, and when the focus is at the pole of the polar coordinate system, the polar equation for a parabola with vertex at the origin becomes \( r = \frac{ep}{1 + e \cos \theta} \), where \( p \) is the semilatus rectum or the distance from the focus to the directrix.\
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In a similar fashion, ellipses and hyperbolas in polar form would also have equations dependent on their eccentricity and directrix but would show different relationships due to their distinct values of eccentricity. For example, for an ellipse, \( e < 1 \), and for a hyperbola, \( e > 1 \). Working with conic sections in polar coordinates often simplifies calculations and visualizations concerning the position of the conics in space, which is why it is a critical concept for students to master, especially in fields such as astronomy, physics, and engineering where these shapes are quite commonplace.