Problem 28 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The polar equation is \(r = \frac{12}{1 - 3\cos\theta}\).

Step 1: Identify the Conic Type

The eccentricity \(e\) of a conic helps to identify its type. If \(e > 1\), the conic is a hyperbola. Since \(e = 3 > 1\), we have a hyperbola.

Step 2: Know the Polar Equation Form for Hyperbolas

The general form of the polar equation for a hyperbola with a focus at the pole is \(r = \frac{ed}{1 - e\cos\theta}\) when the directrix is \(r = -d\sec\theta\).

Step 3: Determine "d" from the Directrix Equation

The given directrix is \(r = -4\sec\theta\), indicating that the value of \(d\) is 4 because the equation is in the form \(r = -d\sec\theta\).

Step 4: Substitute Values into the Polar Equation Form

Substitute \(e = 3\) and \(d = 4\) into the hyperbola polar equation form to get \(r = \frac{3 \cdot 4}{1 - 3\cos\theta}\).

Step 5: Simplify the Polar Equation

Simplifying the substituted values gives the polar equation \(r = \frac{12}{1 - 3\cos\theta}\) for the hyperbola.

Key Concepts

Conic Sections

Conic sections are fascinating shapes created by intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. Each shape has unique properties and equations that describe its geometry.

• **Circle**: Formed when the plane cuts the cone parallel to its base. It's characterized by a constant distance from the center to any point on the edge.
• **Ellipse**: Occurs when the plane cuts through the cone at an angle, creating an oval shape. An ellipse has two foci and the total distance from any point on the ellipse to the foci is constant.
• **Parabola**: Formed when the plane is parallel to a cone's slant. It features a vertex and an axis of symmetry, with all points equidistant from a directrix and focus.
• **Hyperbola**: Created when the plane cuts through both napes of the cone, forming two symmetrical open curves.

Understanding conic sections helps in many branches of science and engineering, as they model paths, orbits, and other phenomena.

Eccentricity

Eccentricity (denoted by \( e \)) is a measure of how much a conic section deviates from being circular. It helps in identifying the type of conic.

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

Larger eccentricity means the conic is more elongated. In the problem, since \( e = 3 \), it confirms that the conic is a hyperbola, making the curves of the section more spread out compared to the symmetry in circles or ellipses.

Directrix

The directrix is a reference line used in the definition of conic sections.It is especially important for parabolas, ellipses, and hyperbolas.

• For a **parabola**, every point is equidistant from the directrix and the focus.
• For an **ellipse** or **hyperbola**, the directrix helps in defining their equations in polar coordinates.

In polar equations, specifying the directrix allows determination of other parameters like distance \( d \). In our exercise, the directrix is given by the equation \( r = -4\sec\theta \), helping to find that \( d = 4 \). Understanding the directrix provides insight into the size and orientation of these conic sections.

Hyperbola

A hyperbola is a type of conic section that forms two mirror-image open curves. It occurs when the plane cuts completely across both nappes of the cone. Hyperbolas have many interesting properties:

• They consist of two branches, each resembling a parabola, but curving away from each other.
• The center of a hyperbola is the midpoint between its vertices (the closest points on each branch).
• The **foci** are points located along the hyperbola's open arms, and they play a crucial role in defining the shape.

In polar coordinates, a hyperbola's equation will show this open nature, as seen in \( r = \frac{12}{1 - 3\cos\theta} \). This equation describes how the radius \( r \) changes with \( \theta \), demonstrating the hyperbolic curve. Understanding hyperbolas is key in many scientific applications, such as orbits and radar detection.