Problem 31 - Thomas Calculus

Question

Exercises \(29-36\) give the eccentricities of conic sections with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$ e=5, \quad y=-6 $$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{30}{1 - 5\sin\theta} \).

Step 1: Understanding the Prompt

We're given the eccentricity, \( e = 5 \), and the directrix equation, \( y = -6 \), for a conic section. Our task is to determine the polar equation of this conic.

Step 2: Identifying the Conic Type

Since the eccentricity \( e \) is greater than 1, this conic section is a hyperbola.

Step 3: Polar Equation Formula

The general formula for a conic in polar form with a directrix perpendicular to the polar axis is \( r = \frac{ed}{1 - e\sin\theta} \), where \( d \) is the distance from the pole to the directrix.

Step 4: Determining the Directrix Distance

Since the directrix is \( y = -6 \), the distance \( d \) from the pole (origin) to this directrix is \( 6 \).

Step 5: Substituting Values

Using \( e = 5 \) and \( d = 6 \), substitute into the polar equation formula: \( r = \frac{5 \times 6}{1 - 5\sin\theta} = \frac{30}{1 - 5\sin\theta} \).

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points in a plane, using a distance and an angle rather than the traditional x and y coordinates. Instead of defining a point's position by experts in a Cartesian grid, it is defined by:

• the distance from a fixed point known as the pole (similar to the origin in Cartesian coordinates),
• and the angle from a fixed direction, typically the positive x-axis, called the polar axis.

This system is particularly useful for problems involving rotational symmetry and is widely used in fields like physics and engineering. The point is often expressed as \(r, \theta\), where \(r\) is the radius, and \(\theta\) is the angle measured in radians. The formula connects polar coordinates to Cartesian coordinates through the relationships: \[ x = r \cos \theta \quad \text{and} \quad y = r \sin \theta.\] This transformation allows us to switch between coordinate systems based on the problem at hand.

Eccentricity

Eccentricity is a parameter that determines the shape of a conic section. It is a numerical measure that tells us how much the conic deviates from being circular. Here are the primary classifications based on eccentricity:

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

In the given exercise, the eccentricity is \(e = 5\), which is greater than 1. Therefore, this informs us that the conic section in question is a hyperbola. Eccentricity impacts the equation of the conic, helping us describe its traits and behavior in various coordinate systems.

Hyperbola

A hyperbola is a type of conic section that occurs when a plane intersects both nappes (the top and bottom halves) of a double cone. Hyperbolas are defined by their eccentricity being greater than 1. Like all conics, hyperbolas have specific geometrical and algebraic properties:

• They consist of two disconnected curves called branches.
• They have a reflective property similar to ellipses but operate with different directrix and foci considerations.

The standard equation of a hyperbola in polar coordinates, when the directrix is perpendicular to the polar axis, is expressed as:\[ r = \frac{ed}{1 - e\sin\theta}\]where \(d\) is the perpendicular distance from the origin to the directrix. This formula derives from the definition involving foci and directrix, showing their eccentric nature.

Directrix

In conic sections, a directrix is a fixed line used in conjunction with the focus to define the conic. The distance to this line in polar coordinates influences the position and shape of the conic:

• In a parabola, the directrix is equidistant from any point on the curve to the focus.
• In an ellipse and hyperbola, the ratio of the distance to the focus versus the distance to the directrix is the eccentricity.

In the exercise at hand, the directrix is given by the line \(y = -6\). For polar coordinates involving this directrix, the distance from the pole (origin) to the directrix is important; hence in the polar equation for the conic, this distance is input as \(d = 6\). This allows us to create a precise mathematical model of the hyperbola, reflecting its unique properties and orientation.