Problem 28 - Essential Calculus

Question

In Exercises \(27-38,\) find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) \(\frac{\text { Conic }}{\text { Parabola }} \quad \frac{\text { Eccentricity }}{e=1} \quad \frac{\text { Directrix }}{x=1}\)

Step-by-Step Solution

Answer

The polar equation for the conic with its focus at the pole is \(r = \frac{1}{1 + \cos(\theta)}\).

Step 1: Analyze given details

Firstly, the given necessary details are: the conic section is a parabola, the eccentricity (\(e\)) equals 1, and the equation for the directrix is \(x=1\), given in rectangular coordinate form.

Step 2: Recall the polar equation of a conic

A conic section's polar equation with focus at the pole and directrix \(x = d\) is obtained as \(r = \frac{ed}{1 + e\cos(\theta)}\) where \(e\) is the eccentricity, \(d\) is the distance from the origin to the directrix, and \(r\) and \(\theta\) are the polar coordinates. Since this is a parabola, \(e=1\). The equation becomes \(r = \frac{d}{1 + \cos(\theta)}\).

Step 3: Substitute the value for directrix into the equation

Since the equation for the directrix is \(x=1\), the distance from the origin to the line \(x=1\) (or the directrix) is \(d=1\). Hence, substituting \(d=1\) back into the polar equation, we obtain \(r = \frac{1}{1 + \cos(\theta)}\). This is the required polar equation for the conic.

Key Concepts

Polar Equations

Polar equations are a way of expressing the locations of points in a plane using the polar coordinate system. This system uses two parameters: the radial distance from a fixed point, called the pole (analogous to the origin in the Cartesian system), and the angle from a fixed direction. Typically, this fixed direction is the positive x-axis.
Let's break that down:

• The radial distance, denoted as \(r\), indicates how far the point is from the pole.
• The angle, denoted as \(\theta\), shows the direction of the point relative to the pole.

In the context of polar conic sections, such as ellipses, hyperbolas, and parabolas, a common form of the polar equation is given by \[r = \frac{ed}{1 + e\cos(\theta)}\]where \(e\) represents the eccentricity of the conic section, and \(d\) is the distance from the pole to the directrix.
By substituting specific values, such as the eccentricity \(e\) and the distance \(d\), we can derive the exact nature and position of the conic section.

Eccentricity

Eccentricity is a fundamental concept when studying conic sections. It is a scalar value that describes the degree to which a conic section deviates from being circular.
For any conic section:

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

In our exercise, we are focusing on a parabola, which means the eccentricity \(e\) is equal to 1.
The eccentricity directly influences the formula: \[r = \frac{1}{1 + \cos(\theta)}\]showing that each conic section has unique geometric properties based on its eccentricity. Understanding eccentricity allows us to precisely identify and graph the conic across different coordinate systems.

Directrix in Polar Coordinates

In the context of conic sections, a directrix is a fixed line used to define and draw the curve.
For polar conic sections, the directrix can affect the curve's shape and orientation. These are often initially given in the familiar rectangular coordinate form, like \(x = 1\), but they must be adjusted to fit within the polar coordinate framework.
Here's how it works:

• The directrix provides a reference line from which distances are measured. The conic is shaped such that its distance to the focus, divided by its distance to the directrix, always equals the eccentricity \(e\).
• For each conic section, the polar coordinate version of the distance to this line (\(d\)) is used in the polar equation \(r = \frac{ed}{1 + e\cos(\theta)}\).

Substituting \(d = 1\), given \(x = 1\), into the equation for our parabola, we derived:\[r = \frac{1}{1 + \cos(\theta)}\]This formula allows us to graphically represent the conic section as it relates to the directrix in polar coordinates.