Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-32: Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$ e=4, \quad r=-3 \csc \theta $$

Step-by-Step Solution

Answer

The polar equation is \(r = \frac{12}{1 + 4\sin \theta}\).

Step 1: Understand the Conic Type Based on Eccentricity

The eccentricity (\(e\)) of the conic section helps us determine its type. If \(e > 1\), the conic is a hyperbola. Since \(e = 4\), this is a hyperbola.

Step 2: Identify the Type of Directrix

The equation is given as \(r = -3 \csc \theta\), which can be rewritten as \(r = -3/\sin \theta\). This indicates that the directrix is a horizontal line parallel to the polar axis at \(y = -3\).

Step 3: Use the Polar Equation Form

For a conic section in polar coordinates with the focus at the pole, the polar equation is given by \(r = \frac{ed}{1 + e\sin \theta}\) (directrix parallel to the polar axis). The value of \(d\) should be positive when the directrix is of the form \(r = -d/\sin \theta\).

Step 4: Substitute Values into the Polar Equation

Given \(e = 4\) and \(d = 3\) (from the directrix), substituting these into the equation yields:\[r = \frac{4 \times 3}{1 + 4\sin \theta} = \frac{12}{1 + 4\sin \theta}.\]This is the polar equation of the hyperbola.

Key Concepts

Conic Sections

Conic sections are a fascinating group of curves that result from intersecting a plane with a cone. They are called conics for short. These sections include circles, ellipses, parabolas, and hyperbolas. Each shape has distinct properties and a unique mathematical representation.

• **Circle**: Every point on a circle is equidistant from its center.
• **Ellipse**: An elongated circle, where the sum of the distances from two fixed points (foci) to every point on the ellipse is constant.
• **Parabola**: A curve where any point is equidistant from a fixed point (focus) and a fixed line (directrix).
• **Hyperbola**: Formed by intersecting a plane with both halves of a cone, resulting in two separate, mirrored curves called branches.

Understanding conic sections is essential in many scientific fields, including astronomy and physics, as they describe paths of celestial bodies and more.

Eccentricity

Eccentricity ( \( e \) ) is a number that describes how "stretched" a conic section is. This value is crucial as it determines the shape and type of the conic section.

• If \( e = 0 \) , the conic is a **circle**.
• If \( 0 < e < 1 \) , the conic is an **ellipse**.
• If \( e = 1 \) , the conic is a **parabola**.
• If \( e > 1 \) , the conic is a **hyperbola**.

Eccentricity explains how much a conic deviates from being circular. In our exercise, with \( e = 4 \) , this indicates a hyperbola, as it exceeds 1. Hyperbolas have the highest eccentricity values among conics.

Hyperbola

A hyperbola is one of the four classic conic sections. It is defined as the set of all points such that the absolute difference of the distances from two fixed points (the foci) is constant.
The structure of a hyperbola consists of two symmetrical curves called branches. These branches open either horizontally or vertically, depending on their orientation. In polar coordinates, given a focus at the pole and an eccentricity greater than 1, the hyperbola's polar equation appears quite distinctive.
In the exercise provided, the eccentricity \( e = 4 \) and the directrix equation led us to derive the polar form of the hyperbola: \( r = \frac{12}{1 + 4\sin \theta} \). This form captures the essence of the hyperbola's geometry relative to the pole.

Directrix

The directrix is a concept central to understanding the geometric definition of conics. It is a reference line used to define and construct a conic section.
For hyperbolas, the distance of any point on the curve from the focus is related to its distance from the directrix by the eccentricity. If the focus is at the pole, as in polar coordinates, the directrix plays a crucial role in determining the position and orientation of the conic.
In common polar equations, such as the one derived for the hyperbola in the exercise, the form involves the directrix: \( r = \frac{ed}{1 + e\sin \theta} \). This expression shows how the distance to the directrix affects the curve. Understanding the role of the directrix helps in visualizing and solving problems related to polar equations.